Hyperspaces of countable compacta

TL;DR

This paper characterizes the topological structure of hyperspaces of countable compact subsets in various metric spaces, revealing homeomorphisms to classical Banach spaces and providing descriptive set-theoretic classifications.

Contribution

It offers a comprehensive topological and descriptive set-theoretic analysis of hyperspaces of countable compacta, including characterizations for specific classes of spaces and new homeomorphism results.

Findings

Hyperspaces of countable compacta are characterized for 0-dimensional Polish spaces.

Certain hyperspaces are homeomorphic to the space c0, including those of intervals and simple closed curves.

The paper provides descriptions of hyperspaces for nondegenerate connected, locally connected Polish spaces.

Abstract

Hyperspaces $\mathcal H(X)$ of all countable compact subsets of a metric space $X$ and $\mathcal A_n(X)$ of infinite compact subsets which have at most $n$ ($n\in\mathbb N$), or finitely many ($n=\omega$) or countably many ($n=\omega+1$) accumulation points are studied. By descriptive set-theoretical methods, we fully characterize them for 0-dimensional, dense-in-itself, Polish spaces and partially for $\sigma$-compact spaces $X$. Using the theory of absorbing sets, we get characterizations of $\mathcal H(X)$, $\mathcal A_\omega(X)$ and $\mathcal A_{\omega+1}(X)$ for nondegenerate connected, locally connected Polish spaces $X$ which are either locally compact or nowhere locally compact. For every $n\in\mathbb N$, we show that if $X$ is an interval or a simple closed curve, $\mathcal A_n(X)$ is homeomorphic to the linear space $c_{0}=\{(x_{i}) \in\mathbb R^{\omega}: \lim x_{i}=0\}$ with the product topology; if $X$ is a Peano continuum and a point $p\in X$ is of order $\ge 2$, then the hyperspace $\mathcal A_1(X,\{p\})$ of all compacta with exactly one accumulation point $p$ also is homeomorphic to $c_{0}$.

Equations

$${\mathcal{A}}_n(X,F):=\{A\in {\mathcal{A}}_n(X):A^{\prime }\subset F\}\cong {\mathbb{Q}}^{\omega }$$

$${\mathcal{A}}_{\omega }(X)\cong {\Sigma }_4,A_{\omega +1}(X)\cong \mathcal{H}(X)\cong \mathcal{H}(\mathbb{I})$$

$${\mathcal{A}}_n(\mathbb{I})\cong {\mathcal{A}}_n((0,1))\cong {\mathcal{A}}_n(S^1)\cong {\Pi }_3\cong c_0:=\{(x_i)\in {\mathbb{R}}^{\omega }:\lim x_i=0\}.$$

$$\{A\in {\mathcal{A}}_n(X):F\subset A\}\cong {\mathcal{A}}_1(X,\{p\})\cong c_0.$$

$$\mathbf{D}:\mathcal{K}(X)\setminus \mathcal{F}(X)\to \mathcal{K}(X),\mathbb{D}(A)=A^{\prime }$$

$${\mathcal{A}}_n(X)={\mathbb{D}}^{-1}({\mathcal{F}}_n(X))\text{ and}{\mathcal{A}}_n(X,F)={\mathbb{D}}^{-1}(F)\cap {\mathcal{A}}_n(X)$$

$$B(x,\epsilon )=\{y\in X:d(y,x)<\epsilon \}\text{mbox}andB[x,\epsilon ]=\{y\in X:d(y,x)\le \epsilon \}$$

$$B(A,\epsilon )=\bigcup _{a\in A}B(a,\epsilon )\text{mbox}andB[A,\epsilon ]=\bigcup _{a\in A}B[a,\epsilon ].$$

$${\mathcal{A}}_{=n}(F,X)=\bigcap _{m\in \mathbb{N}}\bigcup _{A\in [D{]}^{\le n}}\bigcup _{k\in \mathbb{N}}\{K\in {\mathcal{A}}_{=n}(X):K\setminus B[A,\frac{1}{m}]\in [F_k{]}^{\le k}\}$$

$$\begin{gathered} {\mathcal{A}}_{=n}(X)\setminus {\mathcal{A}}_{=n}(X,F)= \\ \bigcap _{k\in \mathbb{N}}\bigcup _{A\in [D{]}^{<n}}\bigcup _{m\in \mathbb{N}}\{K\in {\mathcal{A}}_{=n}(X):K\cap B(F_k,\frac{1}{m})\setminus B[A,\frac{1}{k}]\in [X{]}^{\le m}\} \end{gathered}$$

$$S_{2k}\cap {\mathcal{A}}_{=k}(X)={\mathcal{A}}_{=k}(F,X).$$

$$T_{2k-1}\cap {\mathcal{A}}_{=k}(X)={\mathcal{A}}_{=k}(X)\setminus {\mathcal{A}}_{=k}(X,F).$$

$$T_2={\mathcal{A}}_1(X),T_1={\mathcal{A}}_1(X)\setminus {\mathcal{A}}_1(X,F)={\mathcal{A}}_{=1}(X)\setminus {\mathcal{A}}_{=1}(X,F).$$

$${\mathcal{A}}_n(X,F)=\bigcup _{k=1}^n(T_{2k}\setminus T_{2k-1}),$$

$$Q_1={\mathcal{A}}_1(F,X)\setminus {\mathcal{A}}_1(F)={\mathcal{A}}_{=1}(F,X)\setminus {\mathcal{A}}_{=1}(X,F)$$

$${\mathcal{A}}_n(F)=\bigcup _{k=1}^n(S_{2k}\setminus Q_{2k-1}),$$

$$\chi(n)=\left\{\begin{array}[]{ll}\operatorname{cl}\bigl{\{}2^{-j}+2^{-(j+k)}:1\leq j\leq n-1,k\in\omega\bigr{\}},&\hbox{\text{if $n>1$};}\\ \emptyset,&\hbox{\text{if $n=1$}.}\end{array}\right.$$

$$\psi_{n}(x)=\chi(n)\cup\operatorname{cl}\bigl{(}\bigl{\{}2^{-j}+2^{-(j+k)}x_{2^{j-n}(2k+1)}:j\geq n,k\in\omega\bigr{\}}\bigr{)}\subset A.$$

$${\psi }_n^{-1}({\mathcal{A}}_n(X))=P_3\text{and}{\psi }_1^{-1}({\mathcal{A}}_{\omega }(X))=S_4.$$

$$\mathcal{U}=⟨U_1,\dots ,U_k⟩=\{K\in \mathcal{K}(X):K\subset \bigcup _{i=1}^k{U}_i,(\forall i)U_i\cap K\ne \varnothing \},$$

$$A\cup \{u_1,\dots ,u_k\}\in \mathcal{U}$$

$$\widetilde{{\psi }_n}:\{0,1{\}}^{\mathbb{N}}\to \mathcal{U},\widetilde{{\psi }_n}(x)={\psi }_n(x)\cup \{u_1,\dots ,u_k\}$$

$$(\widetilde{{\psi }_n}{)}^{-1}({\mathcal{A}}_n(X)\cap \mathcal{U})=P_3\text{and}(\widetilde{{\psi }_1}{)}^{-1}({\mathcal{A}}_{\omega }(X)\cap \mathcal{U})=S_4$$

$$(\widetilde{{\psi }_n}{)}^{-1}({\mathcal{A}}_n(X,F)\cap {\mathcal{U}}_F)=P_3\text{and}(\widetilde{{\psi }_1}{)}^{-1}({\mathcal{A}}_{\omega }(X,F)\cap {\mathcal{U}}_F)=S_4$$

$$\delta :X\to {\mathcal{A}}_n(X),\delta (x)=\{x\}\cup K$$

$$h(x)=\bigl{(}h_{n}(\mathrm{pr}_{N_{n}}(x))\bigr{)}_{n\in\omega}.$$

$$T_m=\{x\in \{0,1{\}}^{\mathbb{N}}:(\exists n\ge m)(\forall r\ge n)({\forall }^{\infty }k)(x_{2^r(2k+1)}=0)\}.$$

$$f^{-1}\bigl{(}\mathcal{A}_{\omega+1}(X)\cap\mathcal{U}\bigr{)}=f^{-1}\bigl{(}\mathcal{H}(X)\cap\mathcal{U}\bigr{)}=\mathcal{H}(Q).$$

$${\mathcal{B}}_j=\{A\in \mathcal{K}(C):A\cap (U_j\setminus U_{j+1})\in \mathcal{F}(X_j)\cup \{\varnothing \}\}.$$

$$\Phi :\mathcal{K}(C)\to \mathcal{K}(C)\cup \{\varnothing \},\Phi (A)=A\cap (U_j\setminus U_{j+1})$$

Full text

1. Introduction

All spaces in the paper are metric.

Let $\mathcal{K}(X)$ be the hyperspace of all nonempty compact subsets of $X$ with the Vietoris topology. It is well known that $\mathcal{K}(X)$ shares many basic topological properties of space $X$ like, e.g., completeness, local compactness, compactness, connectedness, local connectedness, dimension 0. Recall also that $\mathcal{K}(X)$ is an absolute neighborhhood retract (ANR) if and only if $X$ is locally continuum-connected and it is an absolute retract (AR) if, additionally, $X$ is connected [11]; if $X$ is nondegenerate noncompact, locally compact, locally connected (connected) then $\mathcal{K}(X)$ is an $\mathbb{I}^{\omega}$-manifold ($\cong\mathbb{I}^{\omega}\setminus\{point\}$) [11]. For a nondegenerate Peano continuum $X$, $\mathcal{K}(X)\cong\mathbb{I}^{\omega}$ (the symbol $\cong$ stands for “homeomorphic to”).

The hyperspace $\mathcal{F}(X)\subset\mathcal{K}(X)$ of all finite subsets of $X$ was also extensively studied for various spaces $X$. Clearly, for the rationals $\mathbb{Q}$, $\mathcal{F}(\mathbb{Q})\cong\mathbb{Q}$. It follows from Lemma 3.9 ([19, Lemma 3.1]) that $\mathcal{F}(\mathbb{R}\setminus\mathbb{Q})\cong\mathbb{Q}\times(\mathbb{R}\setminus\mathbb{Q})$ and $\mathcal{F}(\{0,1\}^{\omega})\cong\mathbb{Q}\times\{0,1\}^{\omega}$. If $X$ is locally path-connected (and connected) then $\mathcal{F}(X)$ is an ANR (AR) which is homotopy dense in $\mathcal{K}(X)$ [15]; for a nondegenerate Peano continuum $X$, $\mathcal{F}(X)\cong[0,1]^{\omega}\setminus(0,1)^{\omega}$ [13].

Another interesting subspace of $\mathcal{K}(X)$ is the hyperspace $\mathcal{H}(X)$ of all nonempty at most countable compacta which seems to have been less recognized. In general, if $X$ is an uncountable Polish space, then $\mathcal{H}(X)$ is $\mathbf{\Pi}^{1}_{1}$-complete [23, Theorem (27.5)] (in such case we will call it the Hurewicz set for $X$). The hyperspace $\mathcal{H}(\mathbb{Q})$ was characterized by H. Michalewski [29] as a first category, zero-dimensional, separable, metrizable space with the property that every nonempty clopen subset is $\mathbf{\Pi}^{1}_{1}$-complete. For $X=\mathbb{I}$, $\mathcal{H}(X)$ was fully topologically characterized by R. Cauty as a $\mathbf{\Pi}^{1}_{1}$-absorbing set, so it is homeomorphic, e.g., to the space of all differentiable functions $\mathbb{I}\to\mathbb{R}$ with the uniform convergence [7].

Denote by $A^{\prime}$ the derived set of all accumulation points of $A\in\mathcal{K}(X)$. In this paper we mainly study the following natural hyperspaces of countable compacta in $X$:

• •

$\mathcal{A}_{n}(X)=\{A\in\mathcal{K}(X):1\leq|A^{\prime}|\leq n\}$ ($n\in\mathbb{N}$),

• •

$\mathcal{A}_{\omega}(X)=\{A\in\mathcal{K}(X):1\leq|A^{\prime}|<\omega\}$.

• •

$\mathcal{H}(X)$ and $\mathcal{A}_{\omega+1}(X)=\{A\in\mathcal{K}(X):1\leq|A^{\prime}|\leq\omega\}=\mathcal{H}(X)\setminus\mathcal{F}(X)$.

Our first task is to evaluate the descriptive complexity of the hyperspaces. Clearly, $\mathcal{A}_{\omega+1}(X)$ is a $G_{\delta}$-subset of $\mathcal{H}(X)$, so whenever $\mathcal{H}(X)$ is absolute coanalytic, $\mathcal{A}_{\omega+1}(X)$ is also such. Evaluations of absolute or exact Borel classes of $\mathcal{A}_{n}(X)$ is more complicated. For any Polish space $X$ without isolated points, $\mathcal{A}_{n}(X)$ is true absolute $F_{\sigma\delta}$ and $\mathcal{A}_{\omega}(X)$ is true absolute $F_{\sigma\delta\sigma}$. For $X=\mathbb{Q}$, $\mathcal{A}_{n}(\mathbb{Q})$ is in the small Borel class $D_{2n}(F_{\sigma\delta})$ in $\mathcal{K}(\mathbb{R})$ but we do not know if it is absolute $F_{\sigma\delta}$.

Next, we characterize $\mathcal{A}_{n}(X)$, $n\in\mathbb{N}$, for any 0-dimensional Polish space which is dense-in-itself (i.e., without isolated points) as the infinite product $\mathbb{Q}^{\omega}$. If $X$ is a 0-dimensional $\sigma$-compact metric space without isolated points then

[TABLE]

for any $F\in\mathcal{K}(X)$ of cardinality $|F|\geq n$. In particular, the hyperspace $\mathcal{A}_{1}(\mathbb{Q},\{q\})=\{A\in\mathcal{A}_{1}(\mathbb{Q}):A^{\prime}=\{q\}\}$ also is homeomorphic to $\mathbb{Q}^{\omega}$. Thus, we get a partial answer to the question asked in [20] if $\mathcal{A}_{1}(\mathbb{R}\setminus\mathbb{Q})$ is homeomorphic to $\mathcal{A}_{1}(\mathbb{Q})$. The full positive answer is equivalent to the $F_{\sigma\delta}$-absoluteness of $\mathcal{A}_{1}(\mathbb{Q})$ which remains an open problem.

We show that for a dense-in-itself, 0-dimensional space $X$, the hyperspace $\mathcal{A}_{\omega}(X)$ is homeomorphic to the standard, everywhere $\mathbf{\Pi}^{0}_{4}$-complete set $S_{4}\subset\{0,1\}^{w}$ in two cases: $X$ a Polish space or a $\sigma$-compact metric space.

If $X$ is a dense-in-itself, 0-dimensional Polish space, then the hyperspaces $A_{\omega+1}(X)$ and $\mathcal{H}(X)$ are homeomorphic to $\mathcal{H}(\mathbb{Q})$.

Studying hyperspaces of compacta of reasonably nice spaces of positive dimensions, we unavoidably enter into infinite-dimensional topology. Here, we intensively employ the theory of absorbing sets. We describe (apparently new) an $F_{\sigma\delta}$-absorber $\Pi_{3}$ and an $F_{\sigma\delta\sigma}$-absorber $\Sigma_{4}$ in the Hilbert cube and our main results in Section 8 are the following characterizations:

[TABLE]

if $X$ is nondegenerate, connected, locally connected and either (1) locally compact or (2) Polish, nowhere locally compact.

One of the simplest examples of type (2)-spaces is $\mathbb{R}^{2}\setminus\mathbb{Q}^{2}$. Other natural examples of such spaces include the set of all “irrational points” of the Sierpiński carpet, infinite countable products of non-compact intervals, Nöbeling or Lipscomb universal spaces of dimension $\geq 1$. In particular, the characterization extends the Cauty’s characterization of $\mathcal{H}(\mathbb{I})$ over all spaces $X$ as in (1) and (2).

The hyperspace $\mathcal{A}_{n}(X)$ is more difficult to handle. In Sections 9 and 10, the following characterizations are obtained:

[TABLE]

Incidentally, the characterizations answer [20, Question 2.17] and a question in [6] if $\mathcal{A}_{n}(S^{1})$ is contractible.

Finally, in Section 11, we show that if $X$ is a Peano continuum with a point $p$ of order $\geq 2$ and $p\in F\in\mathcal{F}(X)$, then

[TABLE]

2. Borel complexity of $\mathcal{A}_{n}(X)$ and $\mathcal{A}_{\omega}(X)$

Let us recall the standard notations of absolute Borel classes of spaces:

• •

for a countable ordinal $\alpha\geq 1$, $\mathbf{\Pi}^{0}_{\alpha}$ is the absolute $\alpha$-th multiplicative class (i.e., $\mathbf{\Pi}^{0}_{1}$ is the class of compact metrizable spaces, $\mathbf{\Pi}^{0}_{2}$ is the class of Polish spaces, etc.);

• •

for $\alpha\geq 2$, $\mathbf{\Sigma}^{0}_{\alpha}$ is the absolute $\alpha$-th additive class (i.e., $\mathbf{\Sigma}^{0}_{2}$ is the class of $\sigma$-compact spaces, $\mathbf{\Sigma}^{0}_{3}$ is the class of absolute $G_{\delta\sigma}$-spaces, etc.).

The class of absolute coanalytic spaces is denoted by $\mathbf{\Pi}^{1}_{1}$.

Let $\Gamma$ be a family of subsets of $X$. For a natural number $n$, let $D_{2n}(\Gamma)$ be the family of sets of type $\bigcup_{k=1}^{n}(A_{2k}\setminus A_{2k-1})$ where $(A_{k})_{k=1}^{n}$ is an increasing sequence of sets from $\Gamma$. For the class $\Gamma$ of $F_{\sigma\delta}$-sets, elements of the small Borel class $D_{2n}(\Gamma)$ will be called sets of difference type $D_{2n}(F_{\sigma\delta})$.

If $\mathcal{P}$ is a topological property, then a subspace $Y\subset Z$ is everywhere (nowhere) $\mathcal{P}$ if every (no) nonempty, relatively open subset of $Y$ has $\mathcal{P}$.

For $E,F\subset X$ and $n\in\mathbb{N}$, denote:

• •

the closed subspace $\mathcal{F}_{n}(X)=\{A\in\mathcal{K}(X):|A|\leq n\}$ of $\mathcal{K}(X)$,

• •

$\mathcal{A}_{=n}(X)=\{A\in\mathcal{K}(X):|A^{\prime}|=n\}$,

• •

$\mathcal{A}_{n}(E,F)=\{A\in\mathcal{A}_{n}(X):A\setminus A^{\prime}\subset E\wedge A^{\prime}\subset F\}.$

• •

$\mathcal{A}_{=n}(E,F)=\{A\in\mathcal{A}_{=n}(X):A\setminus A^{\prime}\subset E\wedge A^{\prime}\subset F\}.$

• •

$\mathcal{K}(X)^{F}=\{K\in\mathcal{K}(X):F\subset K\},\quad\mathcal{A}_{n}(X)^{F}=\mathcal{A}_{n}(X)\cap\mathcal{K}(X)^{F}.$

• •

$\mathcal{K}(X)_{F}=\{K\in\mathcal{K}(X):F\cap K\neq\emptyset\},\quad\mathcal{A}_{n}(X)_{F}=\mathcal{A}_{n}(X)\cap\mathcal{K}(X)_{F}.$

Theorem 2.1**.**

For each $n\in\mathbb{N}$,

1. (1)

$\mathcal{A}_{n}(X)$* and $\mathcal{A}_{n}(X,F)$, for any $F\in\mathcal{K}(X)$, are $F_{\sigma\delta}$-sets in $\mathcal{K}(X)$;* 2. (2)

$\mathcal{A}_{\omega}(X)$* is $F_{\sigma\delta\sigma}$ in $\mathcal{K}(X)$; *

if $X$ is metric separable and $F$ is an $F_{\sigma}$-subset of $X$, then 3. (3)

$\mathcal{A}_{=n}(F,X)$* is of type $F_{\sigma\delta}$ in $\mathcal{A}_{=n}(X)$;* 4. (4)

$\mathcal{A}_{=n}(X,F)$* is of type $G_{\delta\sigma}$ in $\mathcal{A}_{=n}(X)$;* 5. (5)

$\mathcal{A}_{=n}(F)$* is of type $D_{2}(F_{\sigma\delta})$ in $\mathcal{A}_{=n}(X)$;* 6. (6)

$\mathcal{A}_{n}(F,X)$, $\mathcal{A}_{n}(X,F)$ and $\mathcal{A}_{n}(F)$ are of type $D_{2n}(F_{\sigma\delta})$ in $\mathcal{K}(X)$.

Proof.

K. Kuratowski proved in [26] that the derived set map

[TABLE]

is Borel of the second class. Actually, it is convenient to consider $\mathbb{D}$ as a map from the whole $\mathcal{K}(X)$ to the space $\mathcal{K}(X)\cup\{\emptyset\}$ with the isolated point $\{\emptyset\}$ and then a direct proof in [10] of Kuratowski’s theorem shows that the preimage under $\mathbb{D}$ of each closed set in $\mathcal{K}(X)$ is $F_{\sigma\delta}$ in $\mathcal{K}(X)$.

Observe that

[TABLE]

which establishes (1) and yields (2).

Now, fix a metric $d$ generating the topology of $X$. For $x\in X$ and $\varepsilon>0$ denote by

[TABLE]

the open and closed $\varepsilon$-balls centered at $x$. For $A\subseteq X$, let

[TABLE]

Fix a countable dense set $D$ in $X$ and $n\in\mathbb{N}$.

(3). The equality

[TABLE]

witnesses that the sets $\mathcal{A}_{=n}(F,X)$ is of type $F_{\sigma\delta}$ in $\mathcal{A}_{=n}(X)$.

(4). The equality

[TABLE]

shows that the set $\mathcal{A}_{=n}(X)\setminus\mathcal{A}_{=n}(X,F)$ is of type $F_{\sigma\delta}$ in $\mathcal{A}_{=n}(X)$ and hence $\mathcal{A}_{=n}(X,F)$ is of type $G_{\delta\sigma}$ in $\mathcal{A}_{=n}(X)$.

(5). Since $\mathcal{A}_{=n}(F)=\mathcal{A}_{=n}(F,X)\cap\mathcal{A}_{=n}(X,F)$, the set $\mathcal{A}_{=n}(F)$ is of type $D_{2}(F_{\sigma\delta})$ in $\mathcal{A}_{=n}(X)$ according to the preceding two statements.

(6). For every $k\in\mathbb{N}$, choose an $F_{\sigma\delta}$-set $S_{2k}$ in $\mathcal{K}(X)$ such that

[TABLE]

Since $\mathcal{A}_{k}(X)$ and $\mathcal{A}_{k-1}(X)$ are $F_{\sigma\delta}$-sets in $\mathcal{K}(X)$, we can assume that $\mathcal{A}_{k-1}(X)\subseteq S_{2k}\subseteq\mathcal{A}_{k}(X)$. Put $S_{1}=\mathcal{A}_{0}(X)\setminus\mathcal{A}_{0}(F)$ and $S_{2k-1}=\mathcal{A}_{k-1}(X)$ for $k>1$. Since $\mathcal{A}_{n}(F,X)=\bigcup_{k=1}^{n}(S_{2k}\setminus S_{2k-1})$, the set $\mathcal{A}_{n}(F,X)$ is of type $D_{2n}(F_{\sigma\delta})$ in $\mathcal{K}(X)$.

For every $k>1$ put $T_{2k}=\mathcal{A}_{k}(X)$ and choose an $F_{\sigma\delta}$-set $T_{2k-1}$ in $\mathcal{K}(X)$ such that

[TABLE]

Since $\mathcal{A}_{k}(X)$ and $\mathcal{A}_{k-1}(X)$ are $F_{\sigma\delta}$-sets in $\mathcal{K}(X)$, we can assume that $\mathcal{A}_{k-1}(X)\subseteq T_{2k-1}\subseteq\mathcal{A}_{k}(X)$. Put also

[TABLE]

Since

[TABLE]

the set $\mathcal{A}_{n}(X,F)$ is of type $D_{2n}(F_{\sigma\delta})$ in $\mathcal{K}(X)$.

Since $\mathcal{A}_{=1}(X)=\mathcal{A}_{1}(X)$ is of type $F_{\sigma\delta}$ in $\mathcal{K}(X)$, the set $\mathcal{A}_{=1}(F,X)\setminus\mathcal{A}_{=1}(X,F)$ is of type $F_{\sigma\delta}$ in $\mathcal{A}_{=1}(X)$ (by statements (3) and (4)) and hence in $\mathcal{K}(X)$. It means that the set

[TABLE]

is of type $F_{\sigma\delta}$ in $\mathcal{K}(X)$. For $k>1$, let $Q_{2k-1}=S_{2k}\cap T_{2k-1}$.

Since

[TABLE]

the set $\mathcal{A}_{n}(F)$ is of type $D_{2n}(F_{\sigma\delta})$ in $\mathcal{K}(X)$. ∎

Corollary 2.2**.**

If $X$ is a Polish space then the Borel classes of sets in (1), (2), (6) of Theorem 2.1 are absolute.

In order to evaluate Borel classes of $\mathcal{A}_{n}(X)$ and $\mathcal{A}_{\omega}(X)$ from below, we will exploit the standard sets $P_{3}$ and $S_{4}$ which are $\mathbf{\Pi}^{0}_{3}$-complete and $\mathbf{\Sigma}^{0}_{4}$-complete, respectively [23, Exercise (23.1), Exercise (23.6)]. They can be represented in slightly different from [23] but equivalent forms:

• •

$P_{3}=\{(x_{i})_{i\in\mathbb{N}}\in\{0,1\}^{\mathbb{N}}:\forall j\in\omega\;\forall^{\infty}k\in\omega\;\;(x_{2^{j}(2k+1)}=0)\}$

• •

$S_{4}=\{(x_{i})_{i\in\mathbb{N}}\in\{0,1\}^{\mathbb{N}}:\forall^{\infty}j\in\omega\forall^{\infty}k\in\omega\,\;\;(x_{2^{j}(2k+1)}=0)\}$.

Theorem 2.3**.**

If $X$ contains a compactum of the Cantor-Bendixson rank $3$, then $\mathcal{A}_{n}(X)$ is not $G_{\delta\sigma}$ in $\mathcal{K}(X)$ and $\mathcal{A}_{\omega}(X)$ is not $G_{\delta\sigma\delta}$ in $\mathcal{K}(X)$. If $X$ is dense-in-itself then

1. (1)

$\mathcal{A}_{n}(X)$* and $\mathcal{A}_{\omega}(X)$ are nowhere $G_{\delta\sigma}$ and nowhere $G_{\delta\sigma\delta}$ in $\mathcal{K}(X)$, respectively,* 2. (2)

If $F\in\mathcal{K}(X)$ and $|F|\geq n$ then $\mathcal{A}_{n}(X,F)$ is nowhere $G_{\delta\sigma}$ in $\mathcal{K}(X)_{F}$ 3. (3)

If $F\in\mathcal{K}(X)$ is infinite then $\mathcal{A}_{\omega}(X,F)$ is nowhere $G_{\delta\sigma\delta}$ in $\mathcal{K}(X)_{F}$.

Proof.

Let the Cantor-Bendixson rank of some $A\in\mathcal{K}(X)$ equals $3$. Without loss of generality we can assume that A=\operatorname{cl}\bigl{\{}2^{-j}+{2^{-(j+k)}}:j,k\in\omega\bigr{\}}. For any $n\in\mathbb{N}$, put

[TABLE]

Define a continuous map $\psi_{n}:\{0,1\}^{\mathbb{N}}\to\mathcal{K}(X)$ by

[TABLE]

One easily checks that

[TABLE]

This guarantees that $\mathcal{A}_{n}(X)$ is not $G_{\delta\sigma}$ and $\mathcal{A}_{\omega}(X)$ is not $G_{\delta\sigma\delta}$.

Now, assume that $X$ has no isolated points. Then every nonempty open subset of $X$ contains a copy of $A$. Consider a basic open set $\mathcal{U}$ in the Vietoris topology in $\mathcal{K}(X)$:

[TABLE]

where $U_{i}$’s are open subsets of $X$ and pick points $u_{i}\in U_{i}$ for $i=1,\dots,k$.

(1). Assume, without loss of generality, that $A\subset U_{1}$. Then

[TABLE]

and the map

[TABLE]

satisfies

[TABLE]

which completes the proof of (1).

(2) and (3). Let $\mathcal{U}_{F}=\langle U_{1},\dots,U_{k}\rangle\cap\mathcal{K}(X)_{F}$. We can assume that $A\subset\bigcup_{i=1}^{k}U_{i}$ and $\{0\}\cup\{2^{-j}:j=1,\dots,n-1\}\subset F$ in case (2) and $\{0\}\cup\{2^{-j}:j\in\mathbb{N}\}\subset F$ in case (3). Then

[TABLE]

in respective cases.

∎

The following general fact can also be observed.

Proposition 2.4**.**

Let $\mathbf{\Gamma}$ be any absolute Borel class containing class $\mathbf{\Sigma}^{0}_{2}$ or any projective class. If an uncountable metrizable separable space $X$ is not in $\mathbf{\Gamma}$, then $\mathcal{A}_{n}(X)$ is not in $\mathbf{\Gamma}$ for each $n\in\mathbb{N}\cup\{\omega\}$.

Proof.

Suppose $\mathcal{A}_{n}(X)$ belongs to $\mathbf{\Gamma}$. Since $X$ is uncountable, it contains infinitely many accumulation points. So, we can find $K\in\mathcal{A}_{n}(X)$. Consider the continuous map

[TABLE]

and observe that the image $\delta(X)$ is closed in $\mathcal{A}_{n}(X)$, hence it is in $\mathbf{\Gamma}$. Then $\delta(X)\setminus\{K\}$ belongs also to $\mathbf{\Gamma}$. Since $\delta\upharpoonright_{X\setminus K}:X\setminus K\to\delta(X)\setminus\{K\}$ is a homeomorphism, the space $X\setminus K$ is in $\mathbf{\Gamma}$ and so is the space $X=(X\setminus K)\cup K$, a contradiction.

∎

3. Hyperspaces $\mathcal{A}_{n}(X)$ for 0-dimensional $X$

In this section, we characterize hyperspaces $\mathcal{A}_{n}(X),$ $n\leq\omega$, for 0-dimensional Polish or $\sigma$-compact spaces without isolated points.

Lemma 3.1**.**

$P_{3}\cong\mathbb{Q}^{\omega}$. In particular, $P_{3}$ is of the first category (in itself) and nowhere $G_{\delta\sigma}$.

Proof.

Represent $\mathbb{N}$ as the countable disjoint union $\mathbb{N}=\bigcup_{n\in\omega}N_{n},$ where $N_{n}=\{2^{n}(2k+1):k\in\omega\}$ and let $\mathrm{pr}_{N_{n}}:\{0,1\}^{\mathbb{N}}\to\{0,1\}^{N_{n}}$ be the projection. Since the set $\{x\in\{0,1\}^{\mathbb{N}}:\forall^{\infty}i\,(x_{i}=0)\}$ is homeomorphic to $\mathbb{Q}$, the sets $Z_{n}=\mathrm{pr}_{N_{n}}(P_{3})$ are also homeomorphic to $\mathbb{Q}$. For each $n\in\omega$, fix a homeomorphism $h_{n}:Z_{n}\to\mathbb{Q}$ and let a homeomorphism $h:P_{3}\to\mathbb{Q}^{\omega}$ be defined by

[TABLE]

∎

Lemma 3.2**.**

The set $S_{4}$ is of the first category (in itself) and strongly homogeneous (i.e., every two nonempty clopen subsets are homeomorphic). In particular, $S_{4}$ is nowhere $G_{\delta\sigma\delta}$.

Proof.

For each $m\in\omega$, put

[TABLE]

Observe that

• •

$S_{4}=\bigcup_{m}T_{m}$,

• •

$T_{m}\subset T_{m+1}$,

• •

$T_{m}$ is closed in $S_{4}$,

• •

$T_{m}$ is nowhere dense in $T_{m+1}$.

It follows that $S_{4}$ is of the first category.

The strong homogeneity follows directly from the definition of $S_{4}$. ∎

The following theorem is due to Steel [31, Theorem 2] and van Engelen [18, Theorem 4.6].

Lemma 3.3**.**

If $\alpha\geq 3$, then any two 0-dimensional, metric separable, first category spaces from the class $\mathbf{\Pi}^{0}_{\alpha}$ ($\mathbf{\Sigma}^{0}_{\alpha}$) which are nowhere $\mathbf{\Sigma}^{0}_{\alpha}$ ($\mathbf{\Pi}^{0}_{\alpha}$, resp.) are homeomorphic.

It was shown in [20] that $\mathcal{A}_{1}(X)$ is of the first category for any second countable topological space $X$. We provide a quick argument for this fact valid for any $\mathcal{A}_{n}(X)$ in the case of a dense-in-itself, metric, separable, 0-dimensional space $X$.

Lemma 3.4**.**

Let $X$ be a dense-in-itself, metric, separable, 0-dimensional space, $n\leq\omega$ and $F\in\mathcal{K}(X)$ be a set of cardinality $\geq n$. Then the hyperspaces $\mathcal{A}_{n}(X)$, $\mathcal{A}_{n}(X,F)$, $\mathcal{A}_{\omega+1}(X)$ and $\mathcal{H}(X)$ are of the first category.

Proof.

Let $\mathcal{A}$ denote any of the hyperspaces. Let $\{B_{1},B_{2},\ldots\}$ be a clopen base in $X$ which is closed under finite unions and $\mathcal{B}_{i,k}$ be the family of all $A$ in $\mathcal{A}$ such that $1\leq|A\setminus B_{i}|\leq k\}.$ Since $X$ has no isolated points, the sets $\mathcal{B}_{i,k}$ are nowhere dense in $\mathcal{A}$. Clearly, $\mathcal{A}$ is the union $\bigcup_{i,k}\mathcal{B}_{i,k}$. ∎

Now, we get the following characterizations.

Theorem 3.5**.**

If $X$ is a dense-in-itself, 0-dimensional Polish space then

1. (1)

$\mathcal{A}_{n}(X)\cong\mathcal{A}_{n}(X,F)\cong P_{3}\cong\mathbb{Q}^{\omega}$* for each $n\in\mathbb{N}$ and $F\in\mathcal{K}(X)$ of cardinality $\geq n$;* 2. (2)

$\mathcal{A}_{\omega}(X)\cong\mathcal{A}_{\omega}(X,F)\cong S_{4}$* for each infinite $F\in\mathcal{K}(X)$;* 3. (3)

$\mathcal{A}_{\omega+1}(X)\cong\mathcal{H}(X)\cong\mathcal{H}(\mathbb{Q})$.

Proof.

Parts (1) and (2) follow from Corollary 2.2, Theorem 2.3 and Lemmas 3.2–3.4.

For part (3), observe that nonempty clopen subsets of $\mathcal{A}_{\omega+1}(X)$ and $\mathcal{H}(X)$ are $\mathbf{\Pi}^{1}_{1}$-complete. To see this, let $\mathcal{U}=\langle U_{1},\ldots,U_{k}\rangle$ be a basic clopen set in $\mathcal{K}(X)$, where $U_{1},\ldots,U_{k}$ are basic clopen subsets of $X$. Take a countable dense subset $Q$ in $\bigcup_{i=1}^{k}U_{i}$ and choose $A_{1}\in\mathcal{A}_{1}(U_{1})$. The set $\mathcal{U}$ is a Polish 0-dimensional space containing the Hurewicz set $\mathcal{H}(Q)\cong\mathcal{H}(\mathbb{Q})$. Now, the continuous map $f:\mathcal{U}\to\mathcal{U},$ $f(A)=A\cup A_{1}$ satisfies

[TABLE]

Thus, by Lemma 3.4 we can apply the Michalewski’s characterization [29].

∎

Theorem 3.6**.**

If $X$ is a 0-dimensional $\sigma$-compact metric space and $F\in\mathcal{K}(X)$, then $\mathcal{A}_{n}(X,F)$ is in $\mathbf{\Pi}^{0}_{3}$ for each $n\in\mathbb{N}$ and $\mathcal{A}_{\omega}(X,F)$ is in $\mathbf{\Sigma}^{0}_{4}$.

Proof.

We can assume that $X$ is contained in the Cantor set $C$. Consider the derived set operator $\mathbb{D}$ on $\mathcal{K}(C)$. The preimage $D^{-1}\left(\mathcal{F}_{n}\right)$ is $F_{\sigma\delta}$ in $\mathcal{K}(C)$ for each $n\in\mathbb{N}$ and $F_{\sigma\delta\sigma}$ for $n=\omega$. Let $C=U_{1}\supsetneqq U_{2}\supsetneqq\dots$ be clopen subsets of $C$ whose intersection is $F$. The sets $X_{j}=X\cap(U_{j}\setminus U_{j+1})$ are $\sigma$-compact. It follows that sets $\mathcal{F}(X_{j})$ are also $\sigma$-compact. Hence, each $\mathcal{F}(X_{j})\cup\{\emptyset\}$ as a subset of a compact space $\mathcal{K}(C)\cup\{\emptyset\}$ is $\sigma$-compact.

Let

[TABLE]

The intersection map

[TABLE]

is continuous [27, Theorems 2 and 3, p.180], consequently,

[TABLE]

is $F_{\sigma}$ in $\mathcal{K}(C)$.

Observe that

[TABLE]

which shows that

[TABLE]

∎

Theorem 3.7**.**

1. (1)

If $X$ is a dense-in-itself, 0-dimensional $\sigma$-compact metric space then $\mathcal{A}_{n}(X,F)\cong\mathbb{Q}^{\omega}$ for any $F\in\mathcal{K}(X)$ of cardinality $|F|\geq n$; 2. (2)

$\mathcal{A}_{w}(X)\cong\mathcal{A}_{w}(X,F)\cong S_{4}$* for any infinite $F\in\mathcal{K}(X)$.*

Proof.

(1). $\mathcal{A}_{n}(X,F)$ is in $\mathbf{\Pi}^{0}_{3}$ by Theorem 3.6. Theorem 2.3 and Lemma 3.4 guarantee that the hyperspace is nowhere $\mathbf{\Sigma}^{0}_{3}$ and of the first category, so Lemma 3.3 applies.

(2). The space $X$ can be considered as an $F_{\sigma}$-subset of the Cantor set $C$. By Theorem 2.1 (6), the hyperspaces $\mathcal{A}_{n}(X)$, $n\in\mathbb{N}$, are $F_{\sigma\delta\sigma}$-subsets of $\mathcal{K}(C)$, so $\mathcal{A}_{w}(X)$ also is $F_{\sigma\delta\sigma}$ in $\mathcal{K}(C)$, hence $\mathcal{A}_{w}(X)$ is in $\mathbf{\Sigma}^{0}_{4}$. $\mathcal{A}_{w}(X,F)$ is in $\mathbf{\Sigma}^{0}_{4}$ by Theorem 3.6. Both hyperspaces are nowhere $\mathbf{\Pi}^{0}_{4}$ (Theorem 2.3) and of the first category (Lemma 3.4), hence they are homeomorphic to $S_{4}$ by Lemma 3.3. ∎

Corollary 3.8**.**

1. (1)

$\mathcal{A}_{n}(\mathbb{Q},F)\cong\mathcal{A}_{n}(\mathbb{R}\setminus\mathbb{Q})\cong\mathcal{A}_{n}(\mathbb{R}\setminus\mathbb{Q},F^{\prime})\cong\mathbb{Q}^{\omega}$* for any $F\in\mathcal{K}(\mathbb{Q})$ and $F^{\prime}\in\mathcal{K}(\mathbb{R}\setminus\mathbb{Q})$ of cardinalities $\geq n$.* 2. (2)

$\mathcal{A}_{w}(\mathbb{Q})\cong\mathcal{A}_{w}(\mathbb{Q},F)\cong\mathcal{A}_{w}(\mathbb{R}\setminus\mathbb{Q})\cong\mathcal{A}_{w}(\mathbb{R}\setminus\mathbb{Q},F^{\prime})\cong S_{4}$* for any infinite $F\in\mathcal{K}(\mathbb{Q})$ and $F^{\prime}\in\mathcal{K}(\mathbb{R}\setminus\mathbb{Q}).$*

In view of Lemma 3.3 and by Theorem 3.5, Theorem 2.3 and Lemma 3.4, the question asked in [20] if $\mathcal{A}_{1}(\mathbb{R}\setminus\mathbb{Q})$ is homeomorphic to $\mathcal{A}_{1}(\mathbb{Q})$ reduces to the problem of the $F_{\sigma\delta}$-absolutness of $\mathcal{A}_{1}(\mathbb{Q})$ (more generally, of $\mathcal{A}_{n}(\mathbb{Q})$); equivalently, whether or not $\mathcal{A}_{1}(\mathbb{Q})$ is $F_{\sigma\delta}$ in $\mathcal{K}(C)$, where $C$ is a Cantor set. Aiming at this direction, we observe several facts shedding some light on the structure of $\mathcal{A}_{n}(\mathbb{Q})$.

We use the following lemma due to van Engelen [19, Lemma 3.1].

Lemma 3.9**.**

Let $X$ and $Y$ be 0-dimensional metric separable spaces, $X=\bigcup_{i=1}^{\infty}X_{i}$, $Y=\bigcup_{i=1}^{\infty}Y_{i}$ with $X_{i}$ (resp. $Y_{i}$) closed and nowhere dense in $X$ (resp. $Y$) and let every nonempty clopen subset of $X$ (resp. $Y$) contain a closed nowhere dense copy of each $Y_{i}$ (resp $X_{i}$). Then $X\cong Y$.

Lemma 3.10**.**

Every nonempty clopen subset of $\mathcal{A}_{n}(\mathbb{Q})$ contains a closed copy of $\mathcal{A}_{n}(\mathbb{Q})$ for $n\in\mathbb{N}\cup\{\omega\}$.

Proof.

We can assume that a clopen subset $\mathcal{U}$ of $\mathcal{A}_{n}(\mathbb{Q})$ is of the form $\mathcal{U}=\langle U_{1},\dots U_{m}\rangle$ for nonempty disjoint clopen subsets $U_{i}$ of $\mathbb{Q}$. The set $\mathcal{A}_{n}(U_{1})$ is a closed copy of $\mathcal{A}_{n}(\mathbb{Q})$.

Fix points $u_{i}\in U_{i}$, $i=2,\dots,m$. Then

[TABLE]

∎

By [19, Theorem 4.1] and Lemmas 3.4, 3.10 we get

Proposition 3.11**.**

$\mathcal{A}_{n}(\mathbb{Q})$* is strongly homogeneous and $\mathcal{A}_{n}(\mathbb{Q})\cong\mathcal{A}_{n}(\mathbb{Q})\times\mathbb{Q}$ for $n\in\mathbb{N}\cup\{\omega\}$.*

One can easily see

Lemma 3.12**.**

$\mathcal{A}_{1}(\mathbb{Q})^{\{q\}}=\operatorname{cl}\left(\mathcal{A}_{1}(\mathbb{Q},\{q\}\right)$* (the closure in $\mathcal{A}_{1}(\mathbb{Q})).$*

Proposition 3.13**.**

$\mathcal{A}_{1}(\mathbb{Q})^{\{q\}}\cong\mathcal{A}_{1}(\mathbb{Q})\cong\mathcal{A}_{1}(\mathbb{Q})^{\{q\}}\setminus\mathcal{A}_{1}(\mathbb{Q},\{q\})$* for every $q\in\mathbb{Q}$.*

Proof.

We have:

[TABLE]

Each $\mathcal{A}_{1}(\mathbb{Q})^{\{p,q\}}$ is closed and nowhere dense in $\mathcal{A}_{1}(\mathbb{Q})^{\{q\}}$. Similarly, each $\mathcal{A}_{1}(\mathbb{Q})^{\{p\}}$ is closed and nowhere dense in $\mathcal{A}_{1}(\mathbb{Q})$. If $\mathcal{U}=\langle U_{1},\dots U_{m}\rangle\cap\mathcal{A}_{1}(\mathbb{Q})^{\{q\}}$ is nonempty for nonempty disjoint clopen subsets $U_{i}$ of $\mathbb{Q}$ and $q\in U_{1}$, then, as in (3.1),

[TABLE]

is a closed and nowhere dense copy of $\mathcal{A}_{1}(\mathbb{Q})^{\{p\}}$ in $\mathcal{U}$. Analogously, each nonempty clopen set $\langle U_{1},\dots U_{m}\rangle\cap\mathcal{A}_{1}(\mathbb{Q})$ in $\mathcal{A}_{1}(\mathbb{Q})$ contains a closed nowhere dense copy of $\mathcal{A}_{1}(\mathbb{Q})^{\{p,q\}}$. Now apply Lemma  3.9.

To prove the second equivalence, represent $\mathcal{A}_{1}(\mathbb{Q})^{\{q\}}\setminus\mathcal{A}_{1}(\mathbb{Q},\{q\})$ as a union $\bigcup_{k\in\mathbb{N}}\mathcal{B}_{k}$, where

[TABLE]

One can easily check that each $\mathcal{B}_{k}$ is closed, nowhere dense in $\mathcal{A}_{1}(\mathbb{Q})^{\{q\}}\setminus\mathcal{A}_{1}(\mathbb{Q},\{q\})$ as well as it can be embedded as a closed nowhere dense subset in each nonempty clopen subset of $\mathcal{A}_{1}(\mathbb{Q})$. Conversely, each nonempty clopen subset of $\mathcal{A}_{1}(\mathbb{Q})^{\{q\}}\setminus\mathcal{A}_{1}(\mathbb{Q},\{q\})$ contains a closed nowhere dense copy of $\mathcal{A}_{1}(\mathbb{Q})^{\{p\}}$.

∎

A map $\mathcal{A}_{1}(\mathbb{Q},\{0\})\times\mathbb{Q}\to\mathcal{A}_{1}(\mathbb{Q})$ given by the translation $(A,q)\mapsto A+q$ is a continuous bijection (it is not a homeomorphism, though). Hence, by Theorem 3.7, Lemma 3.12 and Proposition 3.13, we get

Corollary 3.14**.**

$\mathcal{A}_{1}(\mathbb{Q})$* is a one-to-one continuous image of $\mathbb{Q}^{\omega}$. Equivalently, $\operatorname{cl}\left(\mathcal{A}_{1}(\mathbb{Q},\{q\}\right)$ is a one-to-one continuous image of $\mathcal{A}_{1}(\mathbb{Q},\{q\})$.*

4. Preliminaries related to strongly universal and absorbing sets

From now on, all spaces are assumed to be metric separable and all maps continuous.

We recall a basic terminology and facts related to absorbing sets. The reader is referred to [1, 3, 4, 30] for more details.

The standard Hilbert cube $\mathbb{I}^{\omega}$ ($\mathbb{I}=[0,1]$) is considered with the metric

[TABLE]

A map $f:X\to Y$ is approximated arbitrarily closely by maps with property $\mathcal{P}$ if for any open cover $\mathcal{U}$ of $Y$ there is a map $g:X\to Y$ with property $\mathcal{P}$ such that $f$ is $\mathcal{U}$-close to $g$, i.e., for each $x\in X$ there is $U\in\mathcal{U}$ containing $\{f(x),g(x)\}$.

A closed subset $B\subset X$ is a (strong) $Z$-set in $X$ if the identity map of $X$ can be approximated arbitrarily closely by maps $f:X\to X$ such that $B\cap f(X)=\emptyset$ ($B\cap\operatorname{cl}(f(X))=\emptyset$).

An embedding $f:X\to Y$ is a $Z$-embedding if $f(X)$ is a $Z$-set in $Y$. A countable union of (compact) $Z$-sets in $X$ will be called a ($\sigma$-compact) $\sigma Z$-set in $X$.

A subset $A\subset Y$ is homotopy dense in $Y$ if there is a deformation $H:Y\times[0,1]\to Y$ such that $H(Y\times(0,1])\subset A$ (a deformation through $A$).

Fact 4.1**.**

[7, Lemma 2.6]** If $M$ is an ANR, a subset $X\subset M$ is homotopy dense in $M$ and $Z$ is a strong $Z$-set in $M$, then $Z\cap X$ is a strong $Z$-set in $X$.

Let $M$ be an absolute neighborhood retract (ANR). It is known that

• •

$B$ is a $Z$-set in $M$ if and only if $M\setminus B$ is homotopy dense in $M$ (see [32, Corollary 3.3]),

• •

if $M$ is completely metrizable and $B$ is a $\sigma Z$-set in $M$, then $M\setminus B$ is homotopy dense in $M$ ([4, Exercise 3, p. 31]),

• •

if $A$ is homotopy dense in $M$ then $A$ is an ANR (an absolute retract (AR) if $M$ is an AR) (see [30, Theorem 4.1.6]).

A space $X$ has the strong discrete approximation property (SDAP) if any map $f:\bigoplus_{n\in\mathbb{N}}\mathbb{I}^{n}\to X$ from the topological sum of finite-dimensional cubes can be approximated arbitrarily closely by maps $g:\bigoplus_{n\in\mathbb{N}}\mathbb{I}^{n}\to X$ such that the family $\{g(\mathbb{I}^{n}):n\in\mathbb{N}\}$ is discrete.

Fact 4.2**.**

[5, Proposition 1.7]**, [4, 1.4.1.] If $M$ is an ANR with SDAP or $M$ is locally compact then every $Z$-set in $M$ is a strong $Z$-set in $M$.

We will also need

Fact 4.3**.**

If $X$ is a homotopy dense subset of a locally compact ANR $M$ and there are $Z$-sets $Z_{i}$ in $M$ such that $X\subset\bigcup_{i\in\mathbb{N}}Z_{i}$, then $X$ has SDAP.

The above fact can be easily derived from Facts 4.2, 4.1 and [4, Theorem 1.4.10] which says that each ANR $X$ that can be represented as a union of countably many strong $Z$-sets in $X$ has SDAP.

The famous Toruńczyk’s theorem [33] says that a space $X$ is an $\\ \mathbb{R}^{\omega}$-manifold if and only if $X$ is a Polish ANR with SDAP.

The following theorem was proved by the first author [4].

Theorem 4.4**.**

A space $X$ is an ANR with SDAP if and only if $X$ is homeomorphic to a homotopy dense subset of an $\mathbb{R}^{\omega}$-manifold.

Let $\mathcal{C}$ be a topological class of spaces. A space $X$ is strongly $\mathcal{C}$-universal if for each $C\in\mathcal{C}$ and closed $B\subset C$, every map $f:C\to X$ which is a $Z$-embedding on $B$ can be approximated arbitrarily closely by $Z$-embeddings $g:C\to X$ such that $g\upharpoonright B=f\upharpoonright B$.

A space $X$ is called $\mathcal{C}$-absorbing if

• •

$X$ is an ANR with SDAP,

• •

$X=\bigcup_{n\in\mathbb{N}}X_{n}$, where each $X_{n}$ is a $Z$-set in $X$ and $X_{n}\in\mathcal{C}$,

• •

$X$ is strongly $\mathcal{C}$-universal.

A fundamental theorem of M. Bestvina and J. Mogilski [5] says that a $\mathcal{C}$-absorbing space is topologically unique up to a homotopy type. In particular,

Theorem 4.5**.**

Any two $\mathcal{C}$-absorbing AR’s are homeomorphic.

It is often more convenient to consider strongly universal pairs and absorbing pairs of spaces.

From now on, $\vec{\mathcal{C}}$ will denote a class of pairs $(K,C)$ such that $K$ is compact, $C\subset K$ and $C\in\mathcal{C}$.

A pair of spaces $(M,X)$ ($X\subset M$) is called

• •

strongly $\vec{\mathcal{C}}$-universal (some authors prefer to say $X$ is strongly $\mathcal{C}$-universal in $M$) if for each pair $(K,C)\in\vec{\mathcal{C}}$ and each closed $B\subset K$ every map $f:K\to M$ which is a $Z$-embedding on $B$ and satisfies $(f\upharpoonright B)^{-1}(X)=B\cap C$ can be approximated arbitrarily closely by $Z$-embeddings $g:K\to M$ such that $g\upharpoonright B=f\upharpoonright B$ and $g^{-1}(X)=C$.

Remarks 4.6*.*

In the above definition,

1. (1)

if $M$ is an ANR, then pairs $(K,C)$ can be replaced by pairs $(\mathbb{I}^{\omega},C)$ [1, Proposition 3.3] ; 2. (2)

if $M$ is an $\mathbb{R}^{\omega}$- or $\mathbb{I}^{\omega}$-manifold, then map $f$ can be replaced by an embedding [4, 1.1.21, 1.1.26]

Proving strong universality of pairs is usually cumbersome. An easier property is the preuniversality which is verified as a first step.

A pair $(M,X)$ is

• •

$\vec{\mathcal{C}}$-preuniversal if for any pair $(K,C)\in\vec{\mathcal{C}}$ there exists a map $f:K\to M$ such that $f^{-1}(X)=C$;

• •

everywhere $\vec{\mathcal{C}}$-preuniversal if for any nonempty open set $U\subseteq X$ and pair $(K,C)\in\vec{\mathcal{C}}$ there exists a map $f:K\to M$ such that $f^{-1}(X)=C$.

Henceforth, we restrict our attention to a Borel or projective class $\mathcal{C}\neq\mathbf{\Pi}^{0}_{2}$ containing all compacta.

We gather several general facts on strongly $\vec{\mathcal{C}}$-universal pairs.

Fact 4.7**.**

(**[1, Corollary 4.4]** If $M$ is an ANR (AR) and $(M,X)$ is strongly $\vec{\mathcal{C}}$-universal, then $X$ and $M\setminus X$ are homotopy dense in $M$ ANR’s (AR’s).

Fact 4.8**.**

[1, Corollary 6.2]** If $M$ is an ANR, $Y\subset M$ is homotopy dense in $M$ and $(Y,X)$ is strongly $\vec{\mathcal{C}}$-universal, then $(M,X)$ is strongly $\vec{\mathcal{C}}$-universal.

Fact 4.9**.**

[1, Lemma 7.1]** If $M$ is an ANR, $(M,X)$ is strongly $\vec{\mathcal{C}}$-universal and $U$ is an nonempty open subset of $E$, then $(U,X\cap U)$ is strongly $\vec{\mathcal{C}}$-universal.

Fact 4.10**.**

[1, Proposition 7.2]** If $M$ is an ANR, $\mathcal{U}$ is an open cover of $M$ and $(U,X\cap U)$ is strongly $\vec{\mathcal{C}}$-universal for every $U\in\mathcal{U}$, then $(M,X)$ is strongly $\vec{\mathcal{C}}$-universal.

Fact 4.11**.**

[1, Theorem 9.5]** If $M$ is an ANR, $(M,X)$ is strongly $\vec{\mathcal{C}}$-universal and $A$ is a $Z$-set in $M$, then $(M,X\cup B)$ is strongly $\vec{\mathcal{C}}$-universal for every subset $B\subset A$.

Fact 4.12**.**

[3, Theorem 3.1]** Suppose $M$ is an ANR, a subset $X\subset M$ has SDAP, $X$ is homotopy dense in $M$ and the pair $(M,X)$ is strongly $\vec{\mathcal{C}}$-absorbing. Then $X$ is strongly $\mathcal{C}$-absorbing.

A pair $(M,X)$ is $\vec{\mathcal{C}}$-absorbing (or $X$ is a $\mathcal{C}$-absorber in $M$), if

• •

$X\in\mathcal{C}$,

• •

$(M,X)$ is strongly $\vec{\mathcal{C}}$-universal,

• •

$X$ is contained in a $\sigma$-compact $\sigma Z$-set in $M$.

A fundamental theorem on absorbing pairs is the following.

Theorem 4.13**.**

[1, Corollary 10.8]** If $M_{i}$ is an $\mathbb{R}^{\omega}$- or $\mathbb{I}^{\omega}$-manifold and pairs $(M_{i},X_{i})$ are $\vec{\mathcal{C}}$-absorbing, $i=1,2$, then $X_{1}\cong X_{2}$ if and only if $X_{1}$ and $X_{2}$ are homotopically equivalent; in particular, if $X_{1}$ and $X_{2}$ are AR’s, then $X_{1}\cong X_{2}$. If $M_{i}$ is an AR for $i=1,2$, then $(M_{1},X_{1})\cong(M_{2},X_{2})$ under a homeomorphism $h$ of pairs (i.e., $h(M_{1})=M_{2}$ and $h(X_{1})=X_{2}$).

Standard $\overrightarrow{\mathbf{\Pi}^{0}_{3}}$-absorbing pairs are $(\mathbb{R}^{\omega},c_{0})$ and $(\mathbb{I}^{\omega},\hat{c_{0}})$, where $\hat{c_{0}}=c_{0}\cap\mathbb{I}^{\omega}$. [16]. More examples of $\overrightarrow{\mathbf{\Pi}^{0}_{3}}$-absorbing pairs can be found in [8, 9, 16, 17, 21, 22, 25, 30].

The Hurewicz set $\mathcal{H}(\mathbb{I})$ is a $\mathbf{\Pi}^{1}_{1}$-absorber in $\mathcal{K}(\mathbb{I})$ [7, 1.4.].

5. Strongly universal sets in Lawson semilattices

A topological semilattice is a topological space $X$ endowed with a continuous commutative, associative operation $*:X\times X\to X$ such that $x*x=x$ for all $x\in X$.

For subsets $A$, $B$ of a semilattice $X$ denote $A*B:=\{a*b:a\in A,b\in B\}$. A subset $A$ of $X$ is a subsemillatice if $A*A\subset A$. A topological semilattice $X$ is called Lawson if it has a base of the topology consisting of subsemilattices. A subsemilattice $A$ of $X$ is a coideal if $(X\setminus A)*X\subset X\setminus A$.

Examples 5.1**.**

1. Natural examples of Lawson semilattices are
2. (1)

Euclidean or Hilbert cubes with $(x_{i})*(y_{i}):=(\max\{x_{i},y_{i}\})$, 3. (2)

Vietoris hyperspaces $\mathcal{K}(X)$ with $A*B:=A\cup B$. 4. (3)

Hyperspaces $\mathcal{F}(X)$, $\mathcal{A}_{\omega}(X)$, $\mathcal{A}_{\omega+1}(X)$ and $\mathcal{H}(X)$ are coideals in $\mathcal{K}(X)$.

A subset $X$ of a space $M$ is called locally path-connected in $M$ if for any point $x\in M$ and neighborhood $U_{x}\subseteq M$ of $x$ there exists a neighborhood $V_{x}\subseteq M$ of $x$ such that for any points $y,z\in V_{x}\cap X$ there exists a continuous map $\gamma:[0,1]\to U_{x}\cap X$ such that $\gamma(0)=y$ and $\gamma(1)=z$. Locally path-connected in $M$ subsets $X$ are also called $LC^{0}$ in $M$.

A space $X$ is locally path-connected ($LC^{0}$) if $X$ is $LC^{0}$ in $X$. If $X$ is locally path-connected in $M$, then $X$ is locally path-connected but not conversely.

The following useful result was proved by W. Kubiś, K. Sakai and M. Yaguchi in [24].

Theorem 5.2**.**

If $X$ is a dense locally path-conected (and connected) subsemilattice in a Lawson semilattice $M$, then $M$ and $X$ are ANR’s (AR’s) and $X$ is homotopy dense in $M$.

The next theorem is an important special case of a more general result recently proved by the first author [2, Theorem 9].

Theorem 5.3**.**

Let $M$ be a Lawson semilattice and $X$ be a dense in $M$ coideal which is $LC^{0}$ in $M$. If our class $\mathcal{C}$ is $\mathbf{\Pi}^{0}_{2}$-hereditary (i.e. for each $C\in\mathcal{C}$, any $G_{\delta}$-subset of $C$ belongs to $\mathcal{C}$), then the following conditions are equivalent:

1. (1)

the pair $(M,X)$ is strongly $\vec{\mathcal{C}}$-universal; 2. (2)

$(M,X)$* is everywhere $\vec{\mathcal{C}}$-preuniversal.*

If $M$ is a Polish space and $X$ has SDAP, then conditions (1) and (2) are equivalent to 3. (3)

$X$* is strongly $\mathcal{C}$-universal.*

6. Two standard Borel absorbers in $\mathbb{I}^{\omega}$

Consider the following subsets of $\mathbb{I}^{\omega}$:

[TABLE]

The sets $\Pi_{3}$ and $\Sigma_{4}$ belong to classes $\Pi_{3}$ and $\mathbf{\Sigma}^{0}_{4}$, respectively, and are connected analogs of $P_{3}$ and $S_{4}$ used in the proof of Theorem 2.3.

As an application of Theorem 5.3, we are going to show that the pairs $(\mathbb{I}^{\omega},\Pi_{3})$, $(\mathbb{I}^{\omega},\Sigma_{4})$ are absorbing for Borel classes $\mathbf{\Pi}^{0}_{3}$ and $\mathbf{\Sigma}^{0}_{4}$, respectively.

Lemma 6.1**.**

1. (1)

The pair $(\mathbb{I}^{\omega},\Sigma_{2})$ is everywhere $\overrightarrow{\mathbf{\Sigma}^{0}_{2}}$-preuniversal; 2. (2)

The pair $(\mathbb{I}^{\omega},\Pi_{3})$ is everywhere $\overrightarrow{\mathbf{\Pi}^{0}_{3}}$-preuniversal; 3. (3)

The pair $(\mathbb{I}^{\omega},\Sigma_{4})$ is everywhere $\overrightarrow{\mathbf{\Sigma}^{0}_{4}}$-preuniversal.

Proof.

We will first prove that the pairs are preuniversal. For every $m\in\omega$ let $\mathrm{pr}_{m}:\mathbb{I}^{\omega}\to\mathbb{I}$, $\mathrm{pr}_{m}:x\mapsto x(m)$, be the coordinate projection.

1. Given a compact metrizable space $K$ and an $F_{\sigma}$-set $C\subset K$, write $C$ as the union $C=\bigcup_{n\in\omega}C_{n}$ of an increasing sequence of closed sets $C_{n}$ in $K$. For every $n\in\omega$ choose a continuous function $f_{n}:K\to\mathbb{I}$ such that $f_{n}^{-1}(0)=C_{n}$. Consider the diagonal product $f=(f_{n})_{n\in\omega}:K\to\mathbb{I}^{\omega}$ and observe that $f^{-1}(\Sigma_{2})=\bigcup_{n\in\omega}C_{n}=C$.

2. Given a compact metrizable space $K$ and an $F_{\sigma\delta}$-set $C\subset K$, write $C$ as the intersection $C=\bigcap_{n\in\omega}C_{n}$ of a decreasing sequence $(C_{n})_{n\in\omega}$ of $F_{\sigma}$-sets in $K$. By the preceding item, for every $n\in\omega$ there exists a continuous map $f_{n}:K\to\mathbb{I}^{\omega}$ such that $f_{n}^{-1}(\Sigma_{2})=C_{n}$. Let $g_{0}:K\to\{0\}\subset\mathbb{I}$ be the constant function and, for every $k\in\mathbb{N}$, let $g_{k}=\mathrm{pr}_{m}\circ f_{n}$ where $n,m\in\omega$ are unique numbers such that $k=2^{n}(2m+1)$. Consider the diagonal product $g=(g_{k})_{k\in\omega}:K\to\mathbb{I}^{\omega}$ and observe that $g^{-1}(\Pi_{3})=\bigcap_{n\in\omega}C_{n}=C$.

3. Let $C$ be any $F_{\sigma\delta\sigma}$-set in a compact metrizable space $K$. By [23, 23.5(i)], there exists a sequence $(C_{n})_{n\in\omega}$ of $F_{\sigma}$-sets $C_{n}$ in $K$ such that $C=\bigcup_{m\in\omega}\bigcap_{n=m}^{\infty}C_{n}$. By the first item, for every $n\in\omega$ there exists a continuous function $f_{n}:K\to\mathbb{I}^{\omega}$ such that $f_{n}^{-1}(\Sigma_{2})=C_{n}$. Let $g_{0}:K\to\{0\}\subset\mathbb{I}$ be the constant function and, for every $k\in\mathbb{N}$, let $g_{k}=\mathrm{pr}_{m}\circ f_{n}$ where $n,m\in\omega$ are unique numbers such that $k=2^{n}(2m+1)$. Consider the diagonal product $g=(g_{k})_{k\in\omega}:K\to\mathbb{I}^{\omega}$ and observe that $g^{-1}(\Sigma_{4})=\bigcup_{m\in\omega}\bigcap_{n=m}^{\infty}C_{n}=C$.

In order to see that the pairs are everywhere preuniversal, fix an open basic set $U=U_{0}\times\dots\times U_{n}\times\mathbb{I}\times\mathbb{I}\dots$ and apply an embedding $h:\mathbb{I}^{\omega}\to U$ which is linear on each of the first $n+1$ coordinates and the identity on the others. Observe that $h$ sends each of the sets $\Sigma_{2}$, $\Pi_{3}$, $\Sigma_{4}$ into itself and use their preuniversality in $\mathbb{I}^{\omega}$.

∎

Recall that $\mathbb{I}^{\omega}$ is a Lawson semilattice (Examples 5.1).

Lemma 6.2**.**

The sets $\Sigma_{2}$, $\Pi_{3}$, $\Sigma_{4}$ are dense coideals in $\mathbb{I}^{\omega}$ and are $LC^{0}$ in $\mathbb{I}^{\omega}$.

Proof.

The first two properties are evident. Let $A\in\{\Sigma_{2},\Pi_{3},\Sigma_{4}\}$. To see that $A$ is $LC^{0}$ in $\mathbb{I}^{\omega}$, consider an open basic set $U=U_{0}\times\dots\times U_{n}\times\mathbb{I}\times\mathbb{I}\dots$ in $\mathbb{I}^{\omega}$, where $U_{i}$ is connected open in $\mathbb{I}$ for each $i\leq n$, and choose arbitrary distinct points $(a_{i})_{i\in\omega},(b_{i})_{i\in\omega}\in U\cap A$. Denote $\Gamma=\{i:a_{i}\neq b_{i}\}$. There is a segment $\gamma(t)=(x_{i}(t))_{i\in\Gamma}\subset\mathbb{I}^{\Gamma}$ from $(a_{i})_{i\in\Gamma}$ to $(b_{i})_{i\in\Gamma}$, $t\in\mathbb{I}$. Put $\bar{\gamma}(t)=(\bar{x}_{i}(t))_{i\in\omega}$, where

[TABLE]

Then $\bar{\gamma}(t)$ is a segment from $(a_{i})_{i\in\omega}$ to $(b_{i})_{i\in\omega}$ in $U\cap A$. ∎

Lemma 6.3**.**

The sets $\Sigma_{2}$, $\Pi_{3}$, $\Sigma_{4}$ are contained in a $\sigma$-compact $\sigma Z$-set in $\mathbb{I}^{\omega}$.

Proof.

This follows from the inclusions

[TABLE]

where X_{i}=\{(x_{n})\in\mathbb{I}^{\omega}:\text{x_{n}=0$for$n\leq i}\,\}, and from the fact that the pseudo-interior $(0,1)^{\omega}$ is homotopy dense in the Hilbert cube $\mathbb{I}^{\omega}$. ∎

Now, Theorem 5.3, Lemmas 6.1, 6.2, 6.3 and Fact 4.3 imply the following corollary.

Corollary 6.4**.**

1. (1)

The pair $(\mathbb{I}^{\omega},\Pi_{3})$ is $\overrightarrow{\mathbf{\Pi}^{0}_{3}}$-absorbing and $\Pi_{3}$ is $\mathbf{\Pi}^{0}_{3}$-absorbing. 2. (2)

The pair $(\mathbb{I}^{\omega},\Sigma_{4})$ is $\overrightarrow{\mathbf{\Sigma}^{0}_{4}}$-absorbing and $\Sigma_{4}$ is $\mathbf{\Sigma}^{0}_{4}$-absorbing.

7. A(N)R properties of $\mathcal{A}_{n}(X)$ ($n\leq\omega+1$) and $\mathcal{H}(X)$

The following lemma is a slight generalization of [15, Lemma 3.2].

Lemma 7.1**.**

If a subspace $X\subset M$ is $LC^{0}$ in $M$, then $\mathcal{F}(X)$ is $LC^{0}$ in $\mathcal{K}(M)$.

Proof.

Let $U$ be an open in $M$ neighborhood of a point $x\in M$. There is an open in $M$ neighborhood $V\subset U$ of $x$ such that any two points $a,b\in V\cap X$ can be joined by a path in $U\cap X$.

Claim 7.1.1*.*

For each finite sets $A,B\subset V\cap X$ there is a path

[TABLE]

Indeed, suppose $|A|\geq|B|$, choose a surjection $s:A\to B$ and paths $\gamma_{a,s(a)}:\mathbb{I}\to\langle U\rangle\cap\mathcal{F}(X)$ such that $\gamma_{a,s(a)}(0)=a$ and $\gamma_{a,s(a)}(1)=s(a)$. Then $\gamma(t):=\bigcup_{a\in A}\gamma_{a,s(a)}(t)$ is the required path.

Now, let $\langle U_{1},\dots,U_{k}\rangle$ be a basic open set in the Vietoris topology in $\mathcal{K}(M)$ and $K\in\langle U_{1},\dots,U_{k}\rangle$.

By the assumption and compactness of $K$, one can find open in $M$ sets $V_{1},\ldots,V_{m}$ such that $K\in\langle V_{1},\dots,V_{m}\rangle\subset\langle U_{1},\dots,U_{k}\rangle,$ each $V_{i}$ is contained in some $U_{j}$ and and any two points $a,b\in V_{i}\cap X$ can be joined by a path in $U_{j}\cap X$ for each $j$ such that $V_{i}\subset U_{j}$. Let $A,B\in\langle V_{1},\dots,V_{m}\rangle\cap\mathcal{F}(X)$. Using Claim 7.1.1, one constructs inductively a path from $A$ to $B$ in $\langle U_{1},\dots,U_{k}\rangle\cap\mathcal{F}(X)$. ∎

Clearly, if $X$ is dense in $M$ then $\mathcal{F}(X)$ is dense in $\mathcal{K}(M)$ and if $X$ is connected, then $\mathcal{F}(X)$ is connected either. Lemma 7.1 and Theorem 5.2 applied to $\mathcal{F}(X)\subset\mathcal{K}(M)$ yield the following lemma.

Lemma 7.2**.**

If $X\subset M$ is dense and $LC^{0}$ in $M$ (and connected), then $\mathcal{F}(X)$ and $\mathcal{K}(X)$ are homotopy dense in $\mathcal{K}(M)$ and the hyperspaces $\mathcal{F}(X)$, $\mathcal{K}(X)$ and $\mathcal{K}(M)$ are ANR’s (AR’s).

Theorem 7.3**.**

If a subspace $X$ of a dense-in-itself space $M$ is dense and $LC^{0}$ in $M$ (and connected), then

1. (1)

$\mathcal{A}_{\omega}(X)$, $\mathcal{A}_{\omega+1}(X)$ and $\mathcal{H}(X)$ are ANR’s (AR’s) which are $LC^{0}$ in $\mathcal{K}(M)$ and homotopy dense in $\mathcal{K}(M)$; 2. (2)

$\mathcal{A}_{n}(X)$* is an ANR (AR) for each $n\in\mathbb{N}$.*

Proof.

First, let us notice that for each $n\in\mathbb{N}\cup\{\omega\}$ the hyperspace $\mathcal{A}_{n}(X)$ is dense in $\mathcal{K}(M)$. This follows easily from the fact that $\mathcal{F}(X)$ is dense in $\mathcal{K}(M)$, $M$ has no isolated points and there are nontrivial paths in $X$ in small neighborhoods of points of $M$.

(1) Let $\mathcal{U}\subset\mathcal{K}(M)$ be an open neighborhood of $K\in\mathcal{K}(M)$. By Lemma 7.1, there is an open $\mathcal{V}\subset\mathcal{U}$ containing $K$ such that any two $F_{1},F_{2}\in\mathcal{F}(X)\cap\mathcal{V}$ can be joined by a path in $\mathcal{F}(X)\cap\mathcal{U}$.

Let $A_{1},A_{2}\in\mathcal{A}_{\omega}(X)\cap\mathcal{V}$. By Lemma 7.2, there is a homotopy

[TABLE]

for each $Y$ and $t>0$. Choose sufficiently small $0<t_{0}<1/2$ such that $H(A_{i},[0,t_{0}])\subset\mathcal{V}$, $i=1,2.$ Put $\gamma_{1}(t)=A_{1}\cup H(A_{1},t)$ for $0\leq t\leq t_{0}$ and let $\gamma:[t_{0},1-t_{0}]\to\mathcal{F}(X)\cap\mathcal{U}$ be a path such that $\gamma(t_{0})=H(A_{1},t_{0})$ and $\gamma(1-t_{0})=H(A_{2},t_{0})$. Then

[TABLE]

is a path in $\mathcal{A}_{\omega}(X)\cap\mathcal{U}$ from $A_{1}$ to $A_{1}\cup A_{2}$. Similarly, there is a path in $\mathcal{A}_{\omega}(X)\cap\mathcal{U}$ from $A_{2}$ to $A_{1}\cup A_{2}$. It means that the hyperspace $\mathcal{A}_{\omega}(X)$ is $LC^{0}$ in $\mathcal{K}(M)$. The proof for $\mathcal{A}_{\omega+1}(X)$ and $\mathcal{H}(X)$ is the same.

Since $\mathcal{A}_{\omega}(X)$ is a dense subsemilattice of $\mathcal{K}(M)$, we conclude by Theorem 5.2 that $\mathcal{A}_{\omega}(X)$ is a homotopy dense in $\mathcal{K}(M)$ ANR, which implies that also $\mathcal{A}_{\omega+1}(X)$ and $\mathcal{H}(X)$ are homotopy dense ANR’s. If $X$ is connected then $\mathcal{K}(M)$ is AR (Lemma 7.2), hence all the hyperspaces are AR’s, as homotopy dense subsets.

(2) Theorem 5.2 is not applicable to hyperspaces $\mathcal{A}_{n}(X)$, $n\in\mathbb{N}$, for they are not subsemilattices of $\mathcal{K}(M)$. Therefore, we provide a more direct argument.

Consider basic open sets $\langle U_{0},U_{1},\dots,U_{k}\rangle$ in the Vietoris topology in $\mathcal{K}(X)$, $k\geq 0,$ where $U_{i}$’s are open path-connected subsets of $X$. The sets $\mathcal{A}_{n}(X)\cap\langle U_{0},U_{1},\dots,U_{k}\rangle$ form an open base in $\mathcal{A}_{n}(X)$ which is closed under finite intersections. We are going to show that each of them is contractible in itself.

Choose points $x_{0}\in U_{0}$ and $s_{i}\in U_{i}$ for each $0<i\leq k$. Let

[TABLE]

For $r\in\mathbb{I}$, let

[TABLE]

The set $\mathcal{F}(X)\cap\langle U_{0},\ldots,U_{k}\rangle$ is dense in the Lawson semilattice $\langle U_{0},\ldots,U_{k}\rangle$ and it is $LC^{0}$ in $\langle U_{0},\ldots,U_{k}\rangle$, by Lemma 7.1, so it is an ANR homotopy dense in $\langle U_{0},\ldots,U_{k}\rangle$, by Theorem 7.3. Hence, there is a homotopy

[TABLE]

i.e.

[TABLE]

The subspace $U=\bigcup_{i=0}^{k}U_{i}$ is locally path-connected and

[TABLE]

is an expansion hyperspace in $U$ in the sense of [15]. Moreover, each element of $\mathcal{E}$ intersects each component of $U$. Therefore $\mathcal{E}$ is an AR [15, Lemma 3.6], so it is contractible. Since $\{x_{0},s_{1},\dots,s_{k}\}\in\mathcal{E}$, there is a homotopy $F:\mathcal{E}\times\mathbb{I}\to\mathcal{E}$ such that

[TABLE]

Define a homotopy

[TABLE]

[TABLE]

Homotopy $G$ is a deformation which contracts $\mathcal{A}_{n}(X)\cap\langle U_{0},U_{1},\dots,U_{k}\rangle$ in itself to the point $S$. In particular, if $X$ is connected then $\mathcal{A}_{n}(X)=\mathcal{A}_{n}(X)\cap\langle X\rangle$ is contractible.

Summarizing: $\mathcal{A}_{n}(X)$ is a locally connected space with an open base closed under finite intersections, each of whose elements is connected and homotopically trivial. It means that $\mathcal{A}_{n}(X)$ is an ANR (see [30, Corollary 4.2.18]); if $X$ is connected then $\mathcal{A}_{n}(X)$ is contractible, hence an AR.

∎

8. Universality and absorbing properties of $\mathcal{A}_{\omega}(X)$, $\mathcal{A}_{\omega+1}(X)$ and $\mathcal{H}(X)$

Lemma 8.1**.**

The pair $(\mathcal{K}(\mathbb{I}),\mathcal{A}_{\omega}(\mathbb{I}))$ is $\overrightarrow{\mathbf{\Sigma}^{0}_{4}}$-preuniversal. The pairs $(\mathcal{K}(\mathbb{I}),\mathcal{A}_{\omega+1}(\mathbb{I}))$ and $(\mathcal{K}(\mathbb{I}),\mathcal{H}(\mathbb{I}))$ are $\overrightarrow{\mathbf{\Pi}^{1}_{1}}$-preuniversal.

Proof.

Consider the map

[TABLE]

Observe that the preimage $\psi^{-1}(\mathcal{A}_{\omega}(\mathbb{I}))=\Sigma_{4}$. Now the strong $\overrightarrow{\mathbf{\Sigma}^{0}_{4}}$-universality of the pair $(\mathbb{I}^{\omega},\Sigma_{4})$ (Corollary 6.4) implies the $\overrightarrow{\mathbf{\Sigma}^{0}_{4}}$-preuniversality of the pair $(\mathcal{K}(\mathbb{I}),\mathcal{A}_{\omega}(\mathbb{I}))$.

The preuniversality of $(\mathcal{K}(\mathbb{I}),\mathcal{H}(\mathbb{I}))$ follows directly from the R. Cauty’s result that the pair is $\overrightarrow{\mathbf{\Pi}^{1}_{1}}$-absorbing [14]. In fact, the construction in [14] shows that $(\mathcal{K}(\mathbb{I}),\mathcal{A}_{\omega+1}(\mathbb{I}))$ is strongly $\overrightarrow{\mathbf{\Pi}^{1}_{1}}$-universal.

∎

Lemma 8.1 implies

Lemma 8.2**.**

Let $X$ be a dense subspace of a space $M$ such that for any non-empty set $U\subset M$ the intersection $U\cap X$ contains a topological copy of the segment $\mathbb{I}$. Then the pair $(\mathcal{K}(M),\mathcal{A}_{\omega}(X))$ is everywhere $\overrightarrow{\mathbf{\Sigma}^{0}_{4}}$-preuniversal and pairs $(\mathcal{K}(M),\mathcal{A}_{\omega+1}(X))$ and $(\mathcal{K}(M),\mathcal{H}(X))$ are everywhere $\overrightarrow{\mathbf{\Pi}^{1}_{1}}$-preuniversal.

Theorem 8.3**.**

If $X$ is a dense subset of a dense-in-itself space $M$ and $X$ is $LC^{0}$ in $M$, then the pair $(\mathcal{K}(M),\mathcal{A}_{\omega}(X))$ is strongly $\overrightarrow{\mathbf{\Sigma}^{0}_{4}}$-universal. The pairs $(\mathcal{K}(M),\mathcal{A}_{\omega+1}(X))$ and $(\mathcal{K}(M),\mathcal{H}(X))$ are strongly $\overrightarrow{\mathbf{\Pi}^{1}_{1}}$-universal.

Proof.

Recall that $\mathcal{A}_{\omega}(X)$, $\mathcal{A}_{\omega+1}(X)$ and $\mathcal{H}(X)$ are dense coideals in the Lawson semilattice $\mathcal{K}(M)$ (Examples 5.1) which is $LC^{0}$ in $\mathcal{K}(M)$ (Theorem 7.3). In view of Lemma 8.2, the conclusion follows from Theorem 5.3. ∎

Theorem 8.4**.**

Let $X$ be a dense $F_{\sigma}$-subset of a dense-in-itself Polish space $M$ and assume $X$ is $LC^{0}$ in $M$.

1. (1)

If $M$ is locally compact, then the pair $(\mathcal{K}(M),\mathcal{A}_{\omega}(X))$ is $\overrightarrow{\mathbf{\Sigma}^{0}_{4}}$-absorbing. The pairs $(\mathcal{K}(M),\mathcal{A}_{\omega+1}(X))$ and $(\mathcal{K}(M),\mathcal{H}(X))$ are $\overrightarrow{\mathbf{\Pi}^{1}_{1}}$-absorbing. 2. (2)

If $M$ is $LC^{0}$ and nowhere locally compact (i.e., no point has a compact neighborhood), then $\mathcal{A}_{\omega}(X)$ is $\mathbf{\Sigma}^{0}_{4}$-absorbing and sets $\mathcal{A}_{\omega+1}(X)$, $\mathcal{H}(X)$ are $\mathbf{\Pi}^{1}_{1}$-absorbing.

Proof.

By Corollary 2.2, the hyperspace $\mathcal{A}_{\omega}(X)$ is an $F_{\sigma\delta\sigma}$-subset of the Polish space $\mathcal{K}(M)$, hence $\mathcal{A}_{\omega}(X)\in\mathbf{\Sigma}^{0}_{4}$. By Theorem 8.3, $(\mathcal{K}(M),\mathcal{A}_{\omega}(X))$ is strongly $\overrightarrow{\mathbf{\Sigma}^{0}_{4}}$-universal. The spaces $\mathcal{A}_{\omega+1}(X)$ and $\mathcal{H}(X)$ belong to class $\mathbf{\Pi}^{1}_{1}$ if $X$ is Polish; if not, then by [23, (33.5)] $\mathcal{K}(X)$ is in $\mathbf{\Pi}^{1}_{1}$ and the hyperspaces $\mathcal{A}_{\omega+1}(X)=\mathcal{A}_{\omega+1}(M)\cap\mathcal{K}(X)$ and $\mathcal{H}(X)=\mathcal{H}(M)\cap\mathcal{K}(X)$ also belong to $\mathbf{\Pi}^{1}_{1}$ as intersections of a Polish space and $\mathbf{\Pi}^{1}_{1}$-sets.

Notice that $\mathcal{A}_{\omega}(X)\subset\mathcal{A}_{\omega+1}(X)\subset\mathcal{H}(X)$. Therefore in case (1), it remains to find a $\sigma$-compact $\sigma Z$-subset of $\mathcal{K}(M)$ that covers $\mathcal{H}(X)$. We may use the following idea due to R. Cauty [7, Lemme 5.6]. We can assume that the metric $\rho$ in $M$ is bounded by 1. The locally path-connected space $X$ admits an equivalent metric

[TABLE]

Define

[TABLE]

Each $Z_{k}$ is a closed subset of $\mathcal{K}(M)$, thus it is $\sigma$-compact. In order to show that the sets are $Z$-sets in $\mathcal{K}(M)$, we apply to them a deformation $H_{t}:\mathcal{K}(M)\to\mathcal{K}(M)$ through $\mathcal{K}(X)$ (it exists by Lemma 7.2) followed by the, so called, expansion deformation

[TABLE]

More precisely, for $t>0$ take a map $f_{t}=E_{t}\circ H_{t}:\mathcal{K}(M)\to\mathcal{K}(X)$. Each $Z_{k}$ contains an isolated point, while $E_{t}(K)$ for $t>0$ has no isolated points. Hence, for sufficiently small $t>0$, $f_{t}$ maps $\mathcal{K}(M)$ into $\mathcal{K}(M)\setminus\bigcup_{k\in\mathbb{N}}Z_{k}$ and approximates the identity map on $\mathcal{K}(M)$. Moreover, since each $A\in\mathcal{H}(X)$ contains an isolated point, we get the desired inclusion $\mathcal{H}(X)\subset\bigcup_{k\in\mathbb{N}}Z_{k}$.

In case (2), $\mathcal{K}(M)$ is an $\mathbb{R}^{\omega}$-manifold (see [12]) and $\mathcal{A}_{\omega}(X),$ $\mathcal{A}_{\omega+1}(X),$ $\mathcal{H}(X)$ being homotopy dense in $\mathcal{K}(M)$, they are ANR’s with SDAP by Theorem 4.4. Moreover, since the pairs $(\mathcal{K}(M),\mathcal{A}_{\omega}(X))$, $(\mathcal{K}(M),\mathcal{A}_{\omega+1}(X))$ and $(\mathcal{K}(M),\mathcal{H}(X))$ are strongly universal in respective classes 8.3, the spaces $\mathcal{A}_{\omega}(X)$, $\mathcal{A}_{\omega+1}(X)$ and $\mathcal{H}(X)$ are universal in the classes by 4.12.

Since $Z$-sets in the $\mathbb{R}^{\omega}$-manifold $\mathcal{K}(M)$ are strong $Z$-sets (Fact 4.2), the sets $Z_{k}\cap\mathcal{A}_{\omega}(X)$, $Z_{k}\cap\mathcal{A}_{\omega+1}(X)$ are $Z$-sets in $\mathcal{A}_{\omega}(X)$, $\mathcal{A}_{\omega+1}(X)$ and $\mathcal{H}(X)$, respectively, by Theorem 7.3 and Fact 4.1. They also belong to the classes. Therefore all sufficient conditions for absorbing sets in the classes are satisfied.

∎

Finally, we get the following characterzations.

Theorem 8.5**.**

$\mathcal{A}_{\omega}(X)\cong\Sigma_{4}$, $A_{\omega+1}(X)\cong\mathcal{H}(X)\cong\mathcal{H}(\mathbb{I})$ in each of the following cases.

1. (1)

$X$* is nondegenerate, connected, locally connected and locally compact (i.e. $X$ is a nondegenerate generalized Peano continuum); if $X$ is compact (i.e., $X$ is a nondegenerate Peano continuum), then*

• •

$(\mathcal{K}(M),\mathcal{A}_{\omega}(X))\cong(\mathbb{I}^{\omega},\Sigma_{4})$,

• •

$(\mathcal{K}(M),\mathcal{A}_{\omega+1}(X))\cong(\mathcal{K}(M),\mathcal{H}(X))\cong(\mathcal{K}(\mathbb{I}),\mathcal{H}(\mathbb{I}))$. 2. (2)

$X$* is nondegenerate, Polish, connected, locally connected and nowhere locally compact.*

Proof.

(1) for noncompact $X$ follows from Theorem 8.4, Corollary 6.4, Theorem 7.3 and Theorem 4.13. In case when $X$ is a nondegenerate Peano continuum, we also use the Curtis-Schori characterization $\mathcal{K}(M)\cong\mathbb{I}^{\omega}$ [14].

(2) follows from Theorem 8.4, Corollary 6.4, Theorem 7.3 and Theorem 4.5.

∎

9. Hyperspaces $\mathcal{A}_{n}(\mathbb{I})$, $n\in\mathbb{N}$

Recall that $\mathcal{K}(\mathbb{I})\cong\mathbb{I}^{\omega}$. It will be more convenient to work with the pair $(\mathcal{K}(\mathbb{J}),\mathcal{A}_{n}(\mathbb{J}))$, where $\mathbb{J}=[-1,1]$.

Lemma 9.1**.**

The pair $(\mathcal{K}(\mathbb{J}),\mathcal{A}_{n}(\mathbb{J}))$ is strongly $\overrightarrow{\mathbf{\Pi}^{0}_{3}}$-universal.

Proof.

We apply an approach developed in [21].

Let $C$ be an $F_{\sigma\delta}$-subset of $\mathbb{I}^{\omega}$, $B$ a closed subset of $\mathbb{I}^{\omega}$, $f:\mathbb{I}^{\omega}\to\mathcal{K}(\mathbb{J})$ an embedding which is a $Z$-embedding on $B$ and $\epsilon>0$. Our goal is to find a $Z$-embedding $g:\mathbb{I}^{\omega}\to\mathcal{K}(\mathbb{J})$ such that $g\upharpoonright B=f\upharpoonright B$, $g^{-1}(\mathcal{A}_{n}(\mathbb{J}))\setminus B=C\setminus B$, and $\operatorname{dist}(f(x),g(x))<\epsilon$ for each $x\in\mathbb{I}^{\omega}$, where $\operatorname{dist}$ denotes the Hausdorff distance in the hyperspace $\mathcal{K}(\mathbb{J})$ (see Remarks 4.6).

The first ingredient in a construction of an approximation $g$ is the embedding $\phi_{n}:\mathbb{I}^{w}\to\mathcal{K}(\mathbb{J}),$

[TABLE]

where $\chi(n)$ is defined in (2.1). The positive part of \phi_{n}\bigl{(}(x_{j})_{j\in\omega}\bigr{)} is responsible for the property $\phi_{n}^{-1}(\mathcal{A}_{n}(\mathbb{J}))=\Pi_{3},$ while the negative one exhibits 1-1 correspondence $\phi_{n}:\mathbb{I}^{\omega}\to\mathcal{K}(\mathbb{J})$.

The pair $(\mathbb{I}^{\omega},\Pi_{3})$ being strongly $\overrightarrow{\mathbf{\Pi}^{0}_{3}}$-universal, there is an embedding $\zeta:\mathbb{I}^{\omega}\to\mathbb{I}^{\omega}$ such that $\zeta^{-1}(\Pi_{3})=C$. Put

[TABLE]

We have

[TABLE]

Next, we need a deformation $H:\mathcal{K}(\mathbb{J})\times\mathbb{I}\to\mathcal{K}(\mathbb{J})$ through finite sets. Deformation $H$ can be easily modified to satisfy

[TABLE]

(see [21, (1-4), p. 183]).

We are going to verify that the embedding $g:\mathbb{I}^{\omega}\to\mathcal{K}(\mathbb{J})$ defined by

[TABLE]

where

[TABLE]

(we use standard operations $\alpha A:=\{\alpha a:a\in A\}$ and $x+A:=\{x+a:a\in A\}$), satisfies the definition of strong $\overrightarrow{\mathbf{\Pi}^{0}_{3}}$-universality of pair $(\mathcal{K}(\mathbb{J}),\mathcal{A}_{n}(\mathbb{J}))$.

Clearly, $g$ is continuous and since $\mu(x)=0$ for $x\in B$, it agrees with $f$ on $B$. It also $\epsilon$-approximates $f$, as

[TABLE]

Mapping $g$ is 1-1 on $B$. So, let $x,y\in\mathbb{I}^{\omega}\setminus B$. Then both numbers $\mu(x)$ and $\mu(y)$ are positive. Suppose $g(x)=g(y)$. It follows that

[TABLE]

hence $\mu(x)=\mu(y)$ and $\zeta(x)_{j}=\zeta(y)_{j}$ for each $j\in\omega$, so $x=y$. Suppose $x\in B$, $y\notin B$ and $g(x)=g(y)$. Then $g(y)=f(x)$. On the other hand,

[TABLE]

by (9.6) and since $f$ is 1-1, a contradiction. Thus, $g$ is 1-1.

It follows from (9.3) that $g^{-1}(\mathcal{A}_{n}(\mathbb{J}))\setminus B=C\setminus B.$

The image $g(\mathbb{I}^{\omega})$ is a $Z$-set in $\mathcal{K}(\mathbb{J})$. Indeed, $g(\mathbb{I}^{\omega}\setminus B)=g(\mathbb{I}^{\omega})\setminus g(B)$ is a $\sigma Z$-set in $\mathcal{K}(\mathbb{J})$ because deformation $H$ through finite sets satisfies

[TABLE]

(since, for each $x\notin B$ and $t>0$, the set $H(g(x),t)$ is finite whereas $g(x)$ is not). Now, $g(B)$ is a $Z$-set, so the union $g(B)\cup g(\mathcal{K}(\mathbb{J})\setminus B)=g(\mathcal{K}(\mathbb{J}))$ is a compact $\sigma Z$-set, hence a $Z$-set in $\mathcal{K}(\mathbb{J})$.

∎

Lemma 9.2**.**

$\mathcal{A}_{n}(\mathbb{J})$* is contained in a $\sigma Z$-set in $\mathcal{K}(\mathbb{J})$.*

Proof.

The $\sigma Z$-set we are looking for was constructed in a more general setting (for generalized Peano continua) in the proof of Case (1) of Theorem 8.4. ∎

Since $\mathcal{A}_{n}(\mathbb{I}))$ is in class $\mathbf{\Pi}^{0}_{3}$, Lemmas 9.1 and 9.2 imply

Theorem 9.3**.**

The pair $(\mathcal{K}(\mathbb{I}),\mathcal{A}_{n}(\mathbb{I}))$ is $\overrightarrow{\mathbf{\Pi}^{0}_{3}}$-absorbing for each $n\in\mathbb{N}$. Consequently, $\mathcal{A}_{n}(\mathbb{I})\cong\Pi_{3}\cong c_{0}\cong\hat{c_{0}}$.

Corollary 9.4**.**

$\mathcal{A}_{n}((0,1))$, $\mathcal{A}_{n}([0,1))$ and $\mathcal{A}_{n}((0,1])$ are also $\mathbf{\Pi}^{0}_{3}$-absorbers in $\mathcal{K}(\mathbb{I})$ for each $n\in\mathbb{N}$. Hence they are all homeomorphic to $c_{0}$.

Proof.

Observe that the set $\mathcal{B}=\{A\in\mathcal{K}(\mathbb{I}):A\cap\{0,1\}\neq\emptyset\}$ is a $Z$-set in $\mathcal{K}(\mathbb{I})$ and $\mathcal{A}_{n}((0,1))=\mathcal{A}_{n}(\mathbb{I})\setminus\mathcal{B}$. It is known from [1, Corollary 9.4] that the difference of an $\mathbf{\Pi}^{0}_{3}$-absorber and a $Z$-set in a Hilbert cube $\mathcal{K}(\mathbb{I})$ is again an $\mathbf{\Pi}^{0}_{3}$-absorber in $\mathcal{K}(\mathbb{I})$. The argument for the remaining intervals is similar. ∎

Remark 9.5*.*

Corollary 9.4 absorbs [20, Theorem 2.4, Corollary 2.5], [6, Theorem 5.1] and provides a positive answer to the question in [20, Question 2.17] of whether or not $\mathcal{S}_{c}([0,1])$ is homeomorphic to $\mathcal{S}_{c}((0,1))$.

Remark 9.6*.*

The above method of showing the strong $\overrightarrow{\mathbf{\Pi}^{0}_{3}}$-universality is specific for $X$ an arc—we continuously select a point from a finite set (the point $\min H(f(x),\mu(x))$ from $H(f(x),\mu(x))$) and such selections are characteristic for arcs [28]. Using Corollary 9.4 and general facts about strongly $\overrightarrow{\mathcal{\mathcal{C}}}$-universal pairs, it is shown in Section 10 that the pair $(\mathcal{K}(S^{1}),\mathcal{A}_{n}(S^{1}))$ is $\overrightarrow{\mathbf{\Pi}^{0}_{3}}$-absorbing. One may ask for what other “nice” spaces $X$ the pair $(\mathcal{K}(X),\mathcal{A}_{n}(X))$ is $\overrightarrow{\mathbf{\Pi}^{0}_{3}}$-absorbing.

10. Hyperspaces $\mathcal{A}_{n}(S^{1})$, $n\in\mathbb{N}$

By $S^{1}$ we denote the unit circle in $\mathbb{R}^{2}$.

Theorem 10.1**.**

The pair $(\mathcal{K}(S^{1}),\mathcal{A}_{n}(S^{1}))$ is $\overrightarrow{\mathbf{\Pi}^{0}_{3}}$-absorbing. Hence, $\mathcal{A}_{n}(S^{1})\cong c_{0}$.

Proof.

The hyperspace $E=\mathcal{K}(S^{m})\setminus\{S^{1}\}$ is an AR. Let $\mathcal{U}$ be its open cover by sets $U_{p}=\mathcal{K}(S^{1}\setminus\{p\})$, $p\in S^{1}$. The pair $(\mathcal{K}(\mathbb{I}),\mathcal{A}_{n}((0,1)^{m})$ is strongly $\overrightarrow{\mathbf{\Pi}^{0}_{3}}$-universal by Corollary 9.4. By Fact 4.9,the pair $(\mathcal{K}((-1,1)),\mathcal{A}_{n}((0,1)))$ is strongly $\overrightarrow{\mathbf{\Pi}^{0}_{3}}$-universal. Let $h:(0,1)\to S^{1}\setminus\{p\}$ be a homeomorphism and $\tilde{h}:\mathcal{K}((0,1))\to\mathcal{K}(S^{1}\setminus\{p\})$ be the induced homeomorphism. Clearly, $\tilde{h}(\mathcal{A}_{n}((0,1)))=\mathcal{A}_{n}(S^{1}\setminus\{p\})$ and the pair $(U_{p},\mathcal{A}_{n}(S^{1}\setminus\{p\}))$ is strongly $\overrightarrow{\mathbf{\Pi}^{0}_{3}}$-universal. Since $\mathcal{A}_{n}(S^{1}\setminus\{p\})=U_{p}\cap\mathcal{A}_{n}(S^{1})$, we infer by Fact 4.10 that the pair $(E,\mathcal{A}_{n}(S^{m}))$ is strongly $\overrightarrow{\mathbf{\Pi}^{0}_{3}}$-universal. The singleton $\{S^{1}\}$ being a $Z$-set in $\mathcal{K}(S^{1})$, $E$ is homotopy dense in $\mathcal{K}(S^{1})$. Then, by Fact 4.8, the hyperspace $(\mathcal{K}(S^{1}),\mathcal{A}_{n}(S^{1}))$ is also strongly $\overrightarrow{\mathbf{\Pi}^{0}_{3}}$-universal.

The proof that $\mathcal{A}_{n}(S^{1})$ is contained in a $\sigma Z$-set in $\mathcal{K}(S^{1})$ is the same as for Lemma 9.2.

∎

11. Hyperspaces $\mathcal{A}_{n}(X)^{F}$

As we have noticed in Remark 9.6, there is an essential obstacle in proving that $\mathcal{A}_{n}(X)$ is an $\mathbf{\Pi}^{0}_{3}$-absorber in $\mathcal{K}(X)$ for nondegenerate Peano continua other than $\mathbb{I}$ and $S^{1}$. The obstacle disappears for hyperspaces $\mathcal{A}_{n}(X)^{F}\subset\mathcal{K}(X)^{F}$ and $\mathcal{A}_{1}(X,\{p\})\subset\mathcal{K}(X)^{\{p\}}$, where $F$ is a fixed finite subset of $X$ which contains a point of order $\geq 2$ and $p$ is a fixed point of order $\geq 2$ (a point $p\in X$ is *of order *$\geq 2$ if there is an arc $L\subset X$ containing $p$ in its combinatorial interior). The latter hyperspace is a natural counterpart of $c_{0}$ whose elements converge to the same number [math].

Theorem 11.1**.**

Suppose $X$ is a Peano continuum, $F\subset X$ is finite and contains a point $p$ of order $\geq 2$, $n\in\mathbb{N}$. Then the pairs $(\mathcal{K}(X)^{F},\mathcal{A}_{n}(X)^{F})$ and $(\mathcal{K}(X)^{\{p\}},\mathcal{A}_{1}(X,\{p\})$ are $\overrightarrow{\mathbf{\Pi}^{0}_{3}}$-absorbing.

Consequently, $\mathcal{A}_{n}(X)^{F}\cong\mathcal{A}_{1}(X,\{p\})\cong c_{0}$.

Proof.

Recall that $\mathcal{K}(X)^{F}$ is a Hilbert cube [14].

Clearly, $\mathcal{A}_{n}(X)^{F}=\mathcal{A}_{n}(X)\cap\mathcal{K}(X)^{F}$ is $F_{\sigma\delta}$ in $\mathcal{K}(X)^{F}$. Also $\mathcal{A}_{1}(X,\{p\})$ is $F_{\sigma\delta}$ in $\mathcal{K}(X)^{\{p\}}$, since it equals the preimage $D^{-1}(\{p\})$, where $D$ is the derived set operator on $\mathcal{K}(X)^{\{p\}}$.

In order to prove the strong $\overrightarrow{\mathbf{\Pi}^{0}_{3}}$-universality, we proceed similarly to the proof of Lemma 9.1.

There is a deformation $H:\mathcal{K}(X)\times[0,1]\to\mathcal{K}(X)$ through finite sets such that $\operatorname{dist}(H(A,t),A)\leq 2t$. If we add $F$ to each $H(A,t)$ we get a continuous deformation $\mathcal{K}(X)^{F}\times[0,1]\to\mathcal{K}(X)^{F}$ through finite sets satisfying $\operatorname{dist}(H(A,t),A)\leq 2t$ for $A\in\mathcal{K}(X)^{F}$. So, we can assume that $H:\mathcal{K}(X)^{F}\times[0,1]\to\mathcal{K}(X)^{F}$ is such. Choose an arc $L\subset X$ containing $p$ in its combinatorial interior. Note that each set $H(A,t)$ contains $p$. To simplify further description, assume without loss of generality that $L=\mathbb{J}=[-1,1]$ and $p=0$. We modify the definition of embedding $\phi_{n}$ from (9.1) in its “negative” part in which the sequence $\{-(2^{-(j+1)}+x_{j}2^{-(j+2)}):j\in\omega\}$ is now replaced with an increasing sequence $l(x)$ obtained in the following way. For any $x=(x_{j})\in\mathbb{I}^{\omega}$, put

[TABLE]

Observe that the sequence $a(x)=(a(x)_{j})$ satisfies

1. (1)

$a(x)$ is strictly increasing and converging to [math], 2. (2)

for each $x,y\in\mathbb{I}^{\omega}$ and $i<j$, vectors

[TABLE]

are not parallel.

Let $x^{\prime}\in\mathbb{I}^{\omega}$ be the sequence $1,x_{1},1,x_{1},x_{2},1,x_{1},x_{2},x_{3},1,\ldots$. Put $l(x)=a(x^{\prime})$. Clearly, $l(x)$ also satisfies conditions (1-2).

Now, let

[TABLE]

Note that, for each $n\in\mathbb{N}$, we have

[TABLE]

Define

[TABLE]

where $\mu(x)$ is defined in (9.5) and $\xi(x)$ is defined by (9.2) with $\psi_{n}$ modified as above. Now, $l(x)$ is responsible for $g(x)$ being 1-1. Indeed, suppose $x,y\in\mathbb{I}^{\omega}\setminus B$ and $g(x)=g(y)$. Then $\mu(x)$ and $\mu(y)$ are positive. If the set

[TABLE]

is non-empty, let $\alpha=\max W$ and notice that, for sufficiently large $k$, say for $k\geq j$, numbers $\mu(x)l(\zeta(x))_{k}$ and $\mu(y)l(\zeta(y))_{k}$ are greater than $\alpha$; if $W=\emptyset$, then put $j=0$. So, we can assume that $\mu(x)l(\zeta(x))_{j}=\mu(y)l(\zeta(y))_{i}$ for some $i\geq j$. Since sequences $\mu(x)l(\zeta(x))$ and $\mu(y)l(\zeta(y))$ are increasing, it follows that also $\mu(x)l(\zeta(x))_{j+1}=\mu(y)l(\zeta(y))_{i+1}$. Thus $i=j$ by property $(2)$. But then $\mu(x)l(\zeta(x))_{k}=\mu(y)l(\zeta(y))_{k}$ for each $k\geq j$. Choose $m\geq j$ such that $(\zeta(x)^{\prime})_{m}=1=(\zeta(y)^{\prime})_{m}$. Then

[TABLE]

which implies $\mu(x)=\mu(y)$. Hence, for each $k\geq j$,

[TABLE]

so $\zeta(x)^{\prime}_{k}=\zeta(y)^{\prime}_{k}.$ Consequently, $\zeta(x)=\zeta(y)$ and $x=y$.

The remaining arguments are exactly the same as in the proofs of Lemmas 9.1 and  9.2 .

∎