A remark on a theorem of Iliadis concerning isometrically containing mappings

TL;DR

This paper provides an alternative proof and refinement of Iliadis's theorem on isometrically containing mappings, also discussing recent related results by Oblakova.

Contribution

It offers a new proof and refinement of Iliadis's theorem, enhancing understanding of isometrically containing mappings and addressing recent related findings.

Findings

Alternative proof of Iliadis's theorem

Refinement of the original result

Discussion of Oblakova's recent results

Abstract

In this paper we give an alternative proof and a refinement of a recent result of S.D.Iliadis concerning isometrically containing mappings. We address also a recent result by A.I.Oblakova.

Full text

1. Introduction

Following Stavros Iliadis [I4], we shall say that a continuous surjection $\;F:X\to Y\;$ between separable metric spaces is isometrically containing for a class $\;\mathscr{F}\;$ of continuous surjections between compact metric spaces if for each $\;f\in\mathscr{F}\;$ there are isometric embeddings

$i:domf\to X$ and $j:ranf\to Y$ of the domain and the range of $f$, respectively, such that $F\circ i=j\circ f$.

Iliadis [I4] proved the following theorem.

Theorem 1 (S.D. Iliadis). For any countable ordinals $\alpha,\beta$ there is a continuous surjection $F:X\to Y$ between complete separable metric spaces with small transfinite dimensions ind$X=\alpha$, ind$Y=\beta$, which is isometrically containing for the class $\mathscr{F}(\alpha,\beta)$ of continuous surjections $f$ between compact metric spaces with ind$(domf)\leq\alpha$ and ind$(ranf)\leq\beta$.

The proof given by Iliadis is based on a rather refined method, developed in his earlier works [I1] - [I3].

The aim of this remark is to provide another proof of this theorem, using some results from [P1], cf. also [PP] and [P2].

This approach gives also some refinement of the Iliadis theorem to the following effect (recall that the dimension of a mapping is the supremum of dimensions of its fibers):

Theorem 2. Given countable ordinals $\alpha$, $\beta$ and an $n$ which is a natural number or $\infty$, there is a continuous surjection $F:X\to Y$ between complete separable metric spaces with ind$X=\alpha$, ind$Y=\beta$, such that dim$F\leq n$ and $F$ is isometrically containing for the class $\mathscr{F}(\alpha,\beta,n)$ of the maps $f\in\mathscr{F}(\alpha,\beta)$ with dim$f\leq n$.

2. Proof of Theorem 2

In the sequel, we shall adopt the notation from the proof of Proposition 5.1.1 in [PP].

Let $M$ be a complete separable metric space isometrically universal for separable metric spaces, let $\mathscr{K}(M)$ be the hyperspace of compact subsets of $M$ equipped with the Vietoris topology, and for a countable ordinal $\gamma$, let $\mathscr{K}_{\gamma}(M)=\{K\in\mathscr{K}(M):\;{\rm ind}K\leq\gamma\}$.

We shall consider also $M$ as a topological subspace of the Hilbert cube $I^{\infty}$, and it should be clear from the context, when we refer to the fixed universal metric on $M$ (which does not extend over $I^{\infty}$) or we just deal with the topology in $M$ (inherited from $I^{\infty}$).

We denote by $C(I^{\infty},I^{\infty})$ the space of continuous maps of $I^{\infty}$ into itself, equipped with the compact - open topology.

Let us fix countable ordinals $\alpha$, $\beta$ and an $n$ which is a natural number or $\infty$.

Let $\Phi_{\alpha}:{\mathbb{N}}^{{\mathbb{N}}}\to\mathscr{K}_{\alpha}(M)$, $\Phi_{\beta}:{\mathbb{N}}^{{\mathbb{N}}}\to\mathscr{K}_{\beta}(M)$ be continuous surjections on the irrationals, considered in [PP], proof of Proposition 5.1.1. Recall that for every $\alpha<\omega_{1}$, the set $G_{\alpha}=\{(t,x):\,x\in\Phi_{\alpha}(t)\}$ is closed in ${\mathbb{N}}^{{\mathbb{N}}}\times M$, ind$G_{\alpha}=\alpha$ and each $K\in\mathscr{K}_{\alpha}(M)$ is isometric with $\{t\}\times\Phi_{\alpha}(t)$, where $\Phi_{\alpha}(t)=K$.

Let

1. (1)

$\mathscr{G}=\{(s,t,f)\in{\mathbb{N}}^{{\mathbb{N}}}\times{\mathbb{N}}^{{\mathbb{N}}}\times C(I^{\infty},I^{\infty}):f(\Phi_{\alpha}(s))=\Phi_{\beta}(t)\hbox{ and }{\rm dim}(f^{-1}(y)\cap\Phi_{\alpha}(s))\leq n\hbox{ for each }y\in\Phi_{\beta}(t)\}$

Using continuity of $\Phi_{\alpha}$ and $\Phi_{\beta}$, and the fact that at most $n$-dimensional compacta form a $G_{\delta}$-set in the hyperspace $\mathscr{K}(I^{\infty})$, cf. [K], $\S$45, IV, Theorem 4, we will show that

1. (2)

$\;\mathscr{G}$ is a $G_{\delta}$-set in the product ${\mathbb{N}}^{{\mathbb{N}}}\times{\mathbb{N}}^{{\mathbb{N}}}\times C(I^{\infty},I^{\infty})$.

Let $\mathscr{E}=\{K\in\mathscr{K}(I^{\infty}):\;{\rm dim}K>n\}$. Then $\mathscr{E}=\bigcup_{m}\mathscr{E}_{m}$, where $\mathscr{E}_{m}$ is closed in the hyperspace. Moreover, the set $\mathscr{E}_{m}^{\star}=\{L\in\mathscr{K}(I^{\infty}):\;L\supset K\hbox{ for some }K\in\mathscr{E}_{m}\}$ is also closed in $\mathscr{K}(I^{\infty})$, contains $\mathscr{E}_{m}$ and is contained in $\mathscr{E}$. Therefore, replacing $\mathscr{E}_{m}$ by $\mathscr{E}_{m}^{\star}$ we can assume that $\mathscr{E}_{m}$ contains all compacta containing some element of $\mathscr{E}_{m}$.

Let us check that, for every $m\in{\mathbb{N}}$,

1. (3)

$\;\mathscr{F}_{m}=\{(s,f,y)\in{\mathbb{N}}^{{\mathbb{N}}}\times C(I^{\infty},I^{\infty})\times I^{\infty}:\;f^{-1}(y)\cap\Phi_{\alpha}(s)\in\mathscr{E}_{m}\}$ is closed in the product ${\mathbb{N}}^{{\mathbb{N}}}\times C(I^{\infty},I^{\infty})\times I^{\infty}$

To that end, consider $(s_{i},f_{i},y_{i})\to(s,f,y)$, $(s_{i},f_{i},y_{i})\in\mathscr{F}_{m}$, and let $C_{i}=f_{i}^{-1}(y_{i})\cap\Phi_{\alpha}(s_{i})\in\mathscr{E}_{m}$. Passing, if necessary, to a subsequence, we can assume that $C_{i}\to C$ in $\mathscr{K}(I^{\infty})$.

We shall check that $C\subset f^{-1}(y)\cap\Phi_{\alpha}(s)$. Let us pick any $a\in C$, and let $a_{i}\to a$, $a_{i}\in C_{i}$. Since $f_{i}(a_{i})=y_{i}$, $f_{i}\to f$, $y_{i}\to y$, we have $f_{i}(a_{i})\to f(a)$, hence $f(a)=y$. Also, since $a_{i}\in\Phi_{\alpha}(s_{i})$ and $\Phi_{\alpha}$ is continuous, $a\in\Phi_{\alpha}(s)$. In effect, $a\in f^{-1}(y)\cap\Phi_{\alpha}(s)$, and $C\subset f^{-1}(y)\cap\Phi_{\alpha}(s)$. Since $C\in\mathscr{E}_{m}$ its superset $f^{-1}(y)\cap\Phi_{\alpha}(s)$ is also in $\mathscr{E}_{m}$, hence $(s,f,y)\in\mathscr{F}_{m}$, which gives us (3).

Now, denoting by $\pi:{\mathbb{N}}^{{\mathbb{N}}}\times C(I^{\infty},I^{\infty})\times I^{\infty}\to{\mathbb{N}}^{{\mathbb{N}}}\times C(I^{\infty},I^{\infty})$ the projection parallel to the compact axis $I^{\infty}$, we conclude that the sets $\pi(\mathscr{F}_{m})$ are closed, and hence the sets

1. (4)

$\;\mathscr{H}_{m}=\left({\mathbb{N}}^{{\mathbb{N}}}\times C(I^{\infty},I^{\infty})\right)\setminus\pi(\mathscr{F}_{m})$

are open in ${\mathbb{N}}^{{\mathbb{N}}}\times C(I^{\infty},I^{\infty})$. Therefore, the set

1. (5)

$\;\bigcap_{m}\mathscr{H}_{m}=\{(s,f)\in{\mathbb{N}}^{{\mathbb{N}}}\times C(I^{\infty},I^{\infty}):\;\hbox{for every }y\in I^{\infty},\hbox{ dim}(f^{-1}(y)\cap\Phi_{\alpha}(s))\leq n\}$

is a $G_{\delta}$-set in ${\mathbb{N}}^{{\mathbb{N}}}\times C(I^{\infty},I^{\infty})$.

Since $\mathscr{L}=\{(s,t,f):\;f(\Phi_{\alpha}(s))=\Phi_{\beta}(t)\}\subset{\mathbb{N}}^{{\mathbb{N}}}\times{\mathbb{N}}^{{\mathbb{N}}}\times C(I^{\infty},I^{\infty})$ is closed, both $\Phi_{\alpha}$, $\Phi_{\beta}$ being continuous, the set $\mathscr{G}=\mathscr{L}\cap\{(s,t,f):\;(s,f)\in\bigcap_{m}\mathscr{H}_{m}\}$ is a $G_{\delta}$-set in ${\mathbb{N}}^{{\mathbb{N}}}\times{\mathbb{N}}^{{\mathbb{N}}}\times C(I^{\infty},I^{\infty})$, which ends the proof of (2).

By (2), there is a continuous surjection of ${\mathbb{N}}^{{\mathbb{N}}}$ onto $\mathscr{G}$,

1. (6)

$\;u\to(s(u),t(u),f_{u})\in\mathscr{G}$, $\,u\in{\mathbb{N}}^{{\mathbb{N}}}$,

and we let

1. (7)

$\;X=\{(u,z):\,z\in\Phi_{\alpha}(s(u))\}\subset{\mathbb{N}}^{{\mathbb{N}}}\times M$,

1. (8)

$\;Y=\{(t,z):\,z\in\Phi_{\beta}(t)\}\subset{\mathbb{N}}^{{\mathbb{N}}}\times M$,

1. (9)

$\;F:X\to Y$, $\,F(u,z)=(t(u),f_{u}(z))$.

The spaces $X$, $Y$ are considered with the metric inherited from the product ${\mathbb{N}}^{{\mathbb{N}}}\times M$, where $M$ is equipped with the universal metric and ${\mathbb{N}}^{{\mathbb{N}}}$ has a standard complete metric.

Then (in notation from [PP]), $Y=G_{\beta}$, hence ind$Y=\beta$, and since the function $(u,z)\to(s(u),z)$ maps $X$ into $G_{\alpha}$, using an observation in [P1], §4, Sublemma 3.2, we check that also ind$X=\alpha$.

The mapping $F$ in (9) is a continuous surjection. For any $(r,w)\in Y$, $F^{-1}(r,w)=\{(u,z)\in{\mathbb{N}}^{{\mathbb{N}}}\times M:\;t(u)=r,\;z\in f_{u}^{-1}(w)\cap\Phi_{\alpha}(s(u))\}$, hence by (1) the projection of $F^{-1}(r,w)$ onto the first coordinate is a perfect map with at most $n$-dimensional fibers. Therefore, dim$F\leq n$.

Finally, let $f:K\to L$ be a continuous between compact spaces with ind$K\leq\alpha$, ind$L\leq\beta$ and dim$f\leq n$. Since $M$ was an isometrically universal space, we can assume that $K$ and $L$ are isometrically embedded in $M$. The function $f:K\to L$ can be extended to a continuous map $\tilde{f}:I^{\infty}\to I^{\infty}$. For $s,t\in{\mathbb{N}}^{{\mathbb{N}}}$ such that $K=\Phi_{\alpha}(s)$, $L=\Phi_{\beta}(t)$, we have $(s,t,\tilde{f})\in\mathscr{G}$, cf. (1), and let $u\in{\mathbb{N}}^{{\mathbb{N}}}$ be such that $s=s(u)$, $t=t(u)$, $\tilde{f}=\tilde{f}_{u}$, cf. (6).

Then, for the isometric identifications $i:K\to\{u\}\times K$ and $j:L\to\{t\}\times L$, we have $F\circ i=j\circ f$, cf. (9).

3. Comment

In [O], Section 4, A.I.Oblakova proved that there exists a Cantor set such that any finite metric space whose diameter does not exceed $1$ and the number of points does not exceed $n$ can be isometrically embedded into it. Using a continuous parametrization of some collections of finite metric spaces, one can refine slightly this result, to the following effect:

Remark. For each natural number $n$ there are Cantor sets $X$ and $Y$ and a continuous surjection $F:X\to Y$ which is isometrically containing for the class $\mathscr{D}_{n}$ of non-expanding surjections between at most $n$-element metric spaces of diameter $\leq 1$.

Proof.

Let us fix a natural number $n$. First we will prove that

1. (10)

there are zero-dimensional compact metric spaces $E,H$ and a continuous surjection $f:E\to H$ which is isometrically containing for the class $\mathscr{D}_{n}$.

Indeed, by a result of Gromov [G], sec.6, there is a compact metric space $Z$ which contains isometrically each metric space of diameter $\leq 1$ containing at most $n$ elements (cf. also [O] and [I2], 9.3, for different proofs).

Let $Z\times Z$ be the metric product and $\pi_{i}:Z\times Z\to Z$, $i=1,2$, projections onto the first and the second coordinate, respectively.

Let $\mathscr{F}$ be the collection of at most $n$-element subsets $T$ of $Z\times Z$ which are graphs of non-expanding surjections from $\pi_{1}(T)$ onto $\pi_{2}(T)$, cf. [PP], (4).

Then $\mathscr{F}$ is compact in the hyperspace of $Z\times Z$ and let $u:2^{{\mathbb{N}}}\to\mathscr{F}$ be a continuous surjection of the Cantor set $2^{{\mathbb{N}}}$ onto $\mathscr{F}$.

We let

1. (11)

$\;E=\{(t,x):\;x\in\pi_{1}(u(t))\}\subset 2^{{\mathbb{N}}}\times Z$,

1. (12)

$H=\{(t,y):\;y\in\pi_{2}(u(t))\}\subset 2^{{\mathbb{N}}}\times Z$,

1. (13)

$f:E\to H$, $\;f(t,x)=(t,y)$, where $(x,y)\in u(t)$.

Notice that continuity of $f$ follows from the fact that its graph $\{\left((t,x),(t,y)\right):\;(x,y)\in u(t)\}$ is a closed subset of the compact product $(2^{{\mathbb{N}}}\times Z)\times(2^{{\mathbb{N}}}\times Z)$.

To see that $f$ is isometrically containing for the class $\mathscr{D}_{n}$, suppose that $g:K\to L$ is a non-expanding surjection between at most $n$-element metric spaces of diameter $\leq 1$. One can assume that $K$ and $L$ are isometrically embedded in $Z$. Since $T=\{(x,g(x)):x\in K\}$ belongs to $\mathscr{F}$, $T=u(t)$ for some $t\in 2^{{\mathbb{N}}}$. Then, for the isometric identifications $i:K\to\{t\}\times K$ and $j:L\to\{t\}\times L$, we have $f\circ i=j\circ g$, cf. (13).

To end the proof of Remark, it suffices to embed $E$ and $H$ into Cantor sets $X$ and $Y$ (by a theorem of Hausdorff, one can extend the metrics on $E$ and $H$ to metrics on $X$ and $Y$, respectively), and to define $F:X\to Y$ as $F=f\circ r$, where $r:X\to E$ is a retraction. Finally, to make sure that $F$ is a surjection, we can always add to $X$ a disjoint copy of $Y$ and let $F$ be the identity on this copy. ∎