Continuous extension of maps between sequential cascades

TL;DR

This paper investigates the properties of contours in sequential cascades, explores contour inversion, and provides a general solution for extending maps between maximal elements to subcascades.

Contribution

It introduces a comprehensive method for continuous extension of maps between maximal elements of sequential cascades to their subcascades.

Findings

Contour inversion for iterated contours is characterized.

A general solution for extending maps between maximal elements is provided.

Topologicity and regularity of convergences are analyzed using contour operations.

Abstract

The contour of a family of filters along a filter is a set-theoretic lower limit. Topologicity and regularity of convergences can be characterized with the aid of the contour operation. Contour inversion is studied, in particular, for iterated contours of sequential cascades. A related problem of continuous extension of maps between maximal elements of sequential cascades to full subcascades is solved in full generality.

Equations

$${\mathrm{Li}}_{\left(n\right)}{\mathcal{F}}_n:={\bigcup }_{k<\omega }{\bigcap }_{n>k}{\mathcal{F}}_n,$$

$$\underset{n<\omega }{\forall }{X}_n\in {\mathcal{F}}_n\cap {\mathcal{G}}_n,$$

$${\mathcal{A}}_1\#{\mathcal{A}}_2\dots \#{\mathcal{A}}_n$$

$${\mathcal{A}}^{\#}:=\left\{H\subset X:\underset{A\in \mathcal{A}}{\forall }H\#A\right\}.$$

$$\mathcal{F}:={\mathrm{Li}}_{\left(n\right)}{\mathcal{F}}_n\text{ and }\mathcal{G}:={\mathrm{Li}}_{\left(k\right)}{\mathcal{G}}_k.$$

$${\mathcal{F}}_n\neg \#{\mathcal{G}}_k,$$

$$\mathcal{F}\neg \#{\mathcal{G}}_k,$$

$${\mathcal{F}}_n\neg \#\mathcal{G}$$

$$\mathcal{F}={\mathrm{Li}}_{{\left(n\right)}_{n>m}}{\mathcal{F}}_n\text{ and }\mathcal{G}={\mathrm{Li}}_{{\left(k\right)}_{k>m}}{\mathcal{G}}_k$$

$$F_{n,k}:=F_{\infty ,k}\text{ and }{G}_{n,k}:=G_{\infty ,k}\text{ if }n>n_k,$$

$$F_{n,k}:=F_{n,\infty }\text{ and }{G}_{n,k}:=G_{n,\infty }\text{ if }k>k_k,$$

$$F_{n,k}:=X_{n,\infty }\text{ and }{G}_{n,k}:=Y_{n,\infty },\text{ otherwise.}$$

$$F_n:={\bigcap }_{k<\omega }{F}_{n,k}\text{ and }{G}_k:={\bigcap }_{n<\omega }{G}_{n,k},$$

$$F:={\bigcup }_{n<\omega }{F}_n\text{ and }G:={\bigcup }_{k<\omega }{G}_k,$$

$$\mathcal{F}$$

$$\mathcal{G}$$

$$\mathcal{H}:={\mathrm{Li}}_{\left(n\right)}{\mathcal{F}}_n={\mathrm{Li}}_{\left(k\right)}{\mathcal{G}}_k,$$

$$\Delta :=\left\{\left(n,k\right):{\mathcal{F}}_n\#{\mathcal{G}}_k\right\}$$

$$\mathcal{H}={\mathrm{Li}}_{\left(n,k\right)}\left({\mathcal{F}}_n\vee {\mathcal{G}}_k\right),$$

$${\mathrm{Li}}_{\left(n,k\right)}\left({\mathcal{F}}_n\vee {\mathcal{G}}_k\right)\ge \mathcal{H}.$$

$$\left\{\left(n,k\right)\in \Delta :n\in A\right\},$$

$${\mathcal{F}}_n={\bigcap }_{f\left(p\right)=n}{\mathcal{H}}_p$$

$${\mathcal{H}}_{n,k}:={\mathcal{F}}_n\vee {\mathcal{G}}_k.$$

$$\beta \left({\bigcup }_{i\in I}{\mathcal{F}}_i\right)={\bigcap }_{i\in I}\beta {\mathcal{F}}_i\text{ and }\beta \left({\bigcap }_{i\in I}{\mathcal{F}}_i\right)={\mathrm{cl}}_{\beta }\left({\bigcup }_{i\in I}\beta {\mathcal{F}}_i\right).$$

$${\mathrm{Ls}}_{\left(n\right)}^{\tau }{A}_n:=\bigcap _{n<\omega }{\mathrm{cl}}_{\tau }(\bigcup _{k>n}{A}_k).$$

$$\beta ({\mathrm{Li}}_{\left(n\right)}{\mathcal{F}}_n)=\bigcap _{n<\omega }{\mathrm{cl}}_{\beta }(\bigcup _{k>n}\beta {\mathcal{F}}_k)={\mathrm{Ls}}_{\left(n\right)}^{\beta }\beta {\mathcal{F}}_n.$$

$$H={\mathrm{Ls}}_{\left(n,k\right)}^{\beta }\left(A_n\cap B_k\right),$$

$${\partial }_{\beta }A={\partial }_{\beta }B,$$

$${\partial }_{\beta }\left\{p_m:m<\omega \right\}={\partial }_{\beta }\left\{q_{2k}:k<\omega \right\}={\partial }_{\beta }\left\{q_k:k<\omega \right\},$$

$${\left(A\right)}_0:=\left\{F\subset X:\mathrm{card}\left(A\setminus F\right)<{\aleph }_0\right\}.$$

$$T\left({\sigma |}_V\right)={T\sigma |}_V.$$

$$l_V(v)=\max \{l_V(s)+1:s⊏v\}.$$

$${\mathrm{sp}}_V\left(v\right):=\left\{r_V\left(s\right):s⊑v\right\}$$

Full text

    2010 Mathematics Subject Classification 54C20, 03E05, 54D55

1. Introduction

00footnotetext: The corresponding author.

The contour $\mathop{\mathrm{L}i}\nolimits_{\left(n\right)}\mathcal{F}_{n}$ of a sequence of filters $\left(\mathcal{F}_{n}\right)_{n}$ on $X$, defined by

[TABLE]

is a special case of the contour operation. This operation is important because it can be used, in complementary ways, to define both diagonal and regular convergences, hence to characterize topologicity and regularity. It was also used by Frolík in [10] to show in ZFC the non-homogeneity of the compactification of a countably infinite set (See Section 8 for more information).

In this paper, we investigate how much information on $\left(\mathcal{F}_{n}\right)_{n}$ can be recovered from the knowledge of $\mathop{\mathrm{L}i}\nolimits_{\left(n\right)}\mathcal{F}_{n},$ that is, a contour inversion problem. In some situations, the contour determines the original sequence almost entirely. For example, it follows from [7, Theorem 3.1] that if $\left(\mathcal{F}_{n}\right)_{n}$ and $\left(\mathcal{G}_{n}\right)_{n}$ are sequences of filters and $\left(X_{n}\right)_{n}$ is a sequence of disjoint sets such that

[TABLE]

then $\mathop{\mathrm{L}i}\nolimits_{\left(n\right)}\mathcal{F}_{n}=\mathop{\mathrm{L}i}\nolimits_{\left(n\right)}\mathcal{G}_{n}$ implies $\mathcal{F}_{n}=\mathcal{G}_{n}$ for almost all $n.$ This is, a particularly simple case, but it has a generalization without the assumption (1.1) (Theorem 3.6). In general, however, the relation $\mathop{\mathrm{L}i}\nolimits_{\left(k\right)}\mathcal{G}_{k}\geq\mathop{\mathrm{L}i}\nolimits_{\left(n\right)}\mathcal{F}_{n}$ does not even imply that $\mathcal{G}_{k}$ and $\mathcal{F}_{n}$ mesh (for the definition, see Section 2 below) for some $\left(n,k\right)$.

The Alternative (Theorem 3.1) is a key to the contour inversion problem. We apply the Alternative to sequential contours, that is, to iterated contours of sequential cascades, to be defined in Section 5. The contour of a sequential cascade is, in fact, the restriction to its maximal elements of the topological modification of the original convergence of the cascade. Therefore, in this case, contour inversion is akin to continuous extension of maps between sequential cascades. We show that if $V,W$ are monotone sequential cascades, then for each continuous map $\varphi:\mathop{\mathrm{e}xt}V\rightarrow\mathop{\mathrm{e}xt}W$ (with respect to the standard topologies of sequential cascades), there exists a full, closed downwards subcascade $T$ of $V$ and a continuous map $f:T\rightarrow W$ such that $\left.f\right|_{\mathop{\mathrm{e}xt}T}=\left.\varphi\right|_{\mathop{\mathrm{e}xt}T}$. This fact was formulated in [7, Theorem 3.3] in a rather special case of injective maps $\varphi$ and repeated in [3] with the same proof that turns out not to be correct.

A topology is called subsequential if it is a subspace of a sequential topology. In [8][3], subsequential spaces were characterized by the following equivalent statements:

1. (1)

A topological space $X$ is subsequential, 2. (2)

$x\in\mathop{\mathrm{c}l}\nolimits_{X}A$ if and only if there exists a sequential contour $\mathcal{F}$ on $A$ such that $x\in\lim\nolimits_{X}\mathcal{F}$, 3. (3)

$A$ is $X$-closed if and only if $\lim\nolimits_{X}\mathcal{F}\subset A$ for each sequential contour $\mathcal{F}$ on $A.$

This shows the importance of sequential contours in the theory of subsequential spaces ([7],[9]). In particular, in [7], a generating role of supercontours, that is, of the suprema $\bigvee_{\alpha<\omega_{1}}\mathcal{F}_{a}$ where $\mathcal{F}_{a}$ is a sequential contour of rank $\alpha,$ was studied. It was shown that if a filter $\mathcal{H}$ generates every subsequential topology, then $\mathcal{H}$ is not a sequential contour. On the other hand, under CH, there exists a supercontour $\mathcal{K}$ such that the class generated by $\mathcal{K}$ strictly includes the class of subsequential topologies.

The continuous extension applies as well in set-theory and logic for some classifications of ultrafilters on natural numbers, for example, to establish a hierarchy of $\omega_{1}$-sequence of disjoint classes of ultrafilters on natural numbers with respect to the level of complication. Specifically, an ultrafilter $u\in P_{\alpha}$ if for each $\beta<\alpha$ there exists a monotone sequential contour $\mathcal{C}$ of rank $\beta$, such that $\mathcal{C}\subset u$ and there is no monotone sequential contour of rank $\alpha$ contained in $u$. It appears that the class $P_{2}$ is precisely that of $P$-points ([17],[19],[16]). The continuous extension (Theorem 7.2) implies that if $u\in P_{\alpha}$ then there is $\gamma\leq\alpha$ such that $f\left(u\right)\in P_{\gamma}$. Monotone sequential cascades are used in [18] to solve Baumgartner’s problem [1], in the first nontrivial case, showing (ZFC) that the class of proper $J_{\omega^{\omega}}$-ultrafilters is empty. It is equally proved in [18] that if there is an arbitrarily long finite $<_{\infty}$-sequence (in Laflamme’s terminology [15]) under $u$, then $u$ is at least a strict $J_{\omega^{\omega+1}}$-ultrafilter.

2. Grill

Recall that families of sets $\mathcal{A}_{1},\mathcal{A}_{2},\ldots,\mathcal{A}_{n}$ mesh, in symbols,

[TABLE]

if $A_{1}\cap A_{2}\cap\ldots\cap A_{n}\neq\varnothing$ for each $A_{1}\in\mathcal{A}_{1},A_{2}\in\mathcal{A}_{2},\ldots$,$A_{n}\in\mathcal{A}_{n}$ [6]. We say that $\mathcal{A},\mathcal{B}$ are disjoint (111Or, dissociated [6].) if $\mathcal{A}\lnot\#\mathcal{B}$. We abridge $\left\{A\right\}\#\mathcal{B}$ to $A\#\mathcal{B}$ and even $\left\{A\right\}\#\left\{B\right\}$ to $A\#B.$ The *grill *$\mathcal{A}^{\#}$ of a family $\mathcal{A}$ of subsets of $X$ is defined by

[TABLE]

A sequence $\left(\mathcal{F}_{n}\right)_{n}$ of filters is called disjoint if $\mathcal{F}_{n}\lnot\#\mathcal{F}_{m}$ for all $n\neq m;$ totally disjoint if there exists a sequence $\left(X_{n}\right)_{n}$ of disjoint sets such that $X_{n}\in\mathcal{F}_{n}$ for each $n<\omega.$

Lemma 2.1**.**

A sequence of filters $\left(\mathcal{F}_{n}\right)_{n}$ is totally disjoint if and only if $\left(\mathcal{F}_{n}\right)_{n}$ is disjoint and $\mathop{\mathrm{L}i}\nolimits_{\left(k\right)}\mathcal{F}_{k}$ is disjoint from $\mathcal{F}_{n}$ for each $n<\omega$.

Proof.

Let $\left(X_{n}\right)_{n}$ be a sequence of disjoint sets such that $X_{n}\in\mathcal{F}_{n}$ for each $n<\omega.$ Then $X_{n}$ and $\bigcup_{n\neq k<\omega}X_{k}$ are disjoint and $\bigcup_{n\neq k<\omega}X_{k}\in\mathop{\mathrm{L}i}\nolimits_{\left(k\right)}\mathcal{F}_{k},$ so that $\mathcal{F}_{n}$ is disjoint from $\mathop{\mathrm{L}i}\nolimits_{\left(k\right)}\mathcal{F}_{k}.$ Conversely, if $\mathcal{F}_{n}$ is disjoint from $\mathop{\mathrm{L}i}\nolimits_{\left(k\right)}\mathcal{F}_{k},$ that is, by definition, there is $k_{n}>n$ such that $X\setminus F_{n}\in\mathcal{F}_{k}$ for each $k\geq k_{n},$ then $F_{n}\in\mathcal{F}_{n}$ and $X\setminus F_{n}\in\mathcal{F}_{k}$ for each $k\geq k_{n}.$ If, moreover, $\left(\mathcal{F}_{n}\right)_{n}$ is disjoint, then also $X\setminus F_{n}\in\mathcal{F}_{k}$ for each $k<k_{n}.$ Define $X_{0}:=F_{0}$ and, inductively, $X_{n}:=F_{n}\setminus\bigcup_{k=0}^{n-1}F_{k}$ for $n>0.$ Then $\left(X_{n}\right)_{n}$ is a sequence of disjoint sets such that $X_{n}\in\mathcal{F}_{n}$ for every $n<\omega.$

3. The Alternative

A relation $A\subset\left\{\left(n,k\right):n<\omega,k<\omega\right\}$ is called transversal if $A$ is infinite, and $\left\{l:\left(n,l\right)\in A\right\}$ and $\left\{m:\left(m,k\right)\in A\right\}$ are at most singletons for each $n,k<\omega.$

Theorem 3.1** (Alternative).**

Let $\left(\mathcal{F}_{n}\right)_{n}$ and $\left(\mathcal{G}_{k}\right)_{k}$ be sequences of filters on a set $X$, and let

[TABLE]

If $\mathcal{F}\#\mathcal{G}$, then one of the following statements holds:

1. A.1.

$\mathcal{F}_{n}\#\mathcal{G}_{k}$* for a transversal set of $\left(n,k\right)$,* 2. A.2.

$\mathcal{F}\#\mathcal{G}_{k}$* for infinitely many $k,$* 3. A.3.

$\mathcal{F}_{n}\#\mathcal{G}$* for infinitely many $n.$*

Proof.

We shall prove the alternative with (A.1) replaced by

1. A.0.

for each $m$ there there exists $\left(n,k\right)$ such that $m<\min\left\{n,k\right\}$ and $\mathcal{F}_{n}\#\mathcal{G}_{k}.$

If none of the cases (A.0)(A.2)(A.3) holds, then there is $m<\omega$ such that

[TABLE]

for each $n,k>m.$ As

[TABLE]

for each $m<\omega,$ we can assume without loss of generality that (3.1)-(3.3) hold for each $n,k<\omega.$

Consequently, by (3.2), for each $k,$ there exist $F_{\infty,k}\in\mathcal{F}$ and $G_{\infty,k}\in\mathcal{G}_{k}$ such that $F_{\infty,k}\cap G_{\infty,k}=\varnothing.$ By the definition of contour, for each $k$ there is $n_{k}>k$ such that $F_{\infty,k}\in\mathcal{F}_{n}$ for $n>n_{k}$. Analogously, for each there exist $F_{n,\infty}\in\mathcal{F}_{n}$ and $G_{n,\infty}\in\mathcal{G}$ such that $F_{n,\infty}\cap G_{n,\infty}=\varnothing.$ By the definition of contour, for each $n$ there exists $k_{n}>n$ such that $G_{n,\infty}\in\mathcal{G}_{k}$ for each $k>k_{n}.$ By (3.1), for each $\left(n,k\right)$ there exist $X_{n,k}\in\mathcal{F}_{n}$ and $Y_{n,k}\in\mathcal{G}_{k}$ such that $X_{n,k}\cap Y_{n,k}=\varnothing.$ Let

[TABLE]

Finally, define

[TABLE]

and notice that the intersections above are finite (!), hence $F_{n}\in\mathcal{F}_{n}$ and $G_{k}\in\mathcal{G}_{k},$ and $F_{n}\cap G_{k}=\varnothing$ for each $\left(n,k\right).$ If

[TABLE]

then $F\in\mathcal{F}$ and $G\in\mathcal{G}$ and $F\cap G=\varnothing$, which completes the proof.

Example 3.2** (Only the first case holds).**

Let $\left(\mathcal{F}_{n}\right)_{n}$ be a totally disjoint and let $\left(\mathcal{G}_{k}\right)_{k}$ be such that $\mathcal{F}_{n}\leq\mathcal{G}_{n}$ for each $n<\omega.$ Then $\mathcal{F}\leq\mathcal{G}$ hence $\mathcal{F}\#\mathcal{G}.$ As $\left(\mathcal{F}_{n}\right)_{n}$ is totally disjoint, $\mathcal{F}_{n}\lnot\#\mathcal{F}$, hence $\mathcal{F}_{n}\lnot\#\mathcal{G}$ thus $\mathcal{G}_{n}\lnot\#\mathcal{F}$ for each $n<\omega.$ Therefore, only (A.1) holds. More precisely, $\mathcal{F}_{n}\#\mathcal{G}_{n},\;\mathcal{F}_{n}\lnot\#\mathcal{G},\;\mathcal{G}_{n}\lnot\#\mathcal{F}$ for each $n<\omega.$

Example 3.3** (Only the last case holds).**

(222A similar example has been communicated to the first author by Tsugunori Nogura as a counter-example to a certain hypothesis.) Let $\left(\mathcal{F}_{n}\right)_{n}$ be a sequence of totally disjoint filters and let $\left(\mathcal{G}_{k}\right)_{k}$ be such that $\mathcal{G}_{k}\geq\mathcal{F}$. Then $\mathcal{F}\lnot\#\mathcal{F}_{n}$ and $\mathcal{G}\geq\mathcal{F}$, hence $\mathcal{G}_{k}\lnot\#\mathcal{F}_{n}$ and $\mathcal{G}\lnot\#\mathcal{F}_{n}$ for each $n,k<\omega.$ Hence, only (A.3) holds. More precisely, $\mathcal{G}_{k}\lnot\#\mathcal{F}_{n},\;\mathcal{G}\lnot\#\mathcal{F}_{n},\;\mathcal{G}_{k}\#\mathcal{F}$ for each $n,k<\omega.$

Of course, by exchanging the roles of the sequences, one gets the case where only (A.2) holds. Here is an example where all the cases hold.

Example 3.4** (All cases hold simultaneously).**

Let $\mathcal{F}_{n}\leq\mathcal{F}_{n+1}\leq\ldots\leq\mathcal{G}_{k}\leq\mathcal{G}_{k+1}.$ Then $\mathcal{F}_{n}\leq\mathcal{G}_{k}$ (hence $\mathcal{F}_{n}\#\mathcal{G}_{k}$) for each $n,k<\omega$. Also $\mathcal{F}\leq\mathcal{G}_{k}$ hence $\mathcal{F}\#\mathcal{G}_{k}$ for each $k<\omega,$ and $\mathcal{F}_{n}\leq\mathcal{G}$ hence $\mathcal{F}_{n}\#\mathcal{G}$ for each $n<\omega,$ and $\mathcal{F}\#\mathcal{G}$.

It turns out that all possible variants of the Alternative can hold. More precisely, if $\varnothing\neq J\subset\left\{1,2,3\right\},$ then there exist sequences of filters $\left(\mathcal{F}_{n}\right)_{n}$ and $\left(\mathcal{G}_{k}\right)_{k}$ such that ($A.j$) holds for $j\in J$ and does not hold for $j\notin J.$ This can be done even so that the conditions hold for all the indices, in the sense that will be made precise below.

Proposition 3.5**.**

If $\varnothing\neq J\subset\left\{1,2,3\right\},$ then there exist sequences of filters $\left(\mathcal{F}_{n}\right)_{n}$ and $\left(\mathcal{G}_{k}\right)_{k}$ such that (A.j) holds for $j\in J$ and does not hold for $j\notin J$ for every $n=k<\omega.$

Proof.

We have seen (Examples 3.3, 3.2 and the subsequent comment) that for each $j\in\left\{1,2,3\right\},$ there exist $\left(\mathcal{F}_{j,n}\right)_{n}$ and $\left(\mathcal{G}_{j,k}\right)_{k}$ on $X_{j}$ such that (A.j) holds for each $n=k<\omega,$ and (A.i) does not hold for every $n,k<\omega$ for $i\neq j.$ We assume that $X_{j}\cap X_{k}=\varnothing$ if $j\neq k$ and $j,k\in J.$ Then

[TABLE]

fulfill the condition.

Here we get a first answer to our contour inversion problem:

Theorem 3.6**.**

If $\left(\mathcal{F}_{n}\right)_{n}$ and $\left(\mathcal{G}_{k}\right)_{k}$ are totally disjoint sequences and if

[TABLE]

then the relation

[TABLE]

is finite-to finite with cofinite domain and range, and

[TABLE]

where $\left(n,k\right)$ denotes the cofinite filter on $\Delta.$ Moreover, $\mathcal{F}_{n}=\bigcap\nolimits_{k\in\Delta n}\left(\mathcal{F}_{n}\vee\mathcal{G}_{k}\right)$ for almost all $n$ and $\mathcal{G}_{k}=\bigcap\nolimits_{n\in\Delta^{-1}k}\left(\mathcal{F}_{n}\vee\mathcal{G}_{k}\right)$ for almost all $k.$

Proof.

As $\left(\mathcal{F}_{n}\right)_{n}$ is totally disjoint, $\mathcal{H}\lnot\#\mathcal{F}_{n}$ for each $n.$ As $\left(\mathcal{G}_{k}\right)_{k}$ is totally disjoint, $\mathcal{H}\lnot\#\mathcal{G}_{k}$ for each $k.$ Therefore, by the Alternative (Theorem 3.1), there is a transversal subset of $\Delta.$ Actually, $\Delta$ itself is finite-to-finite with cofinite domain and range. Indeed, if for some $n$ the set $\Delta n$ were infinite, then $\mathop{\mathrm{L}i}\left\{\mathcal{F}_{n}\vee\mathcal{G}_{k}:k\in\Delta n\right\}\geq\mathcal{F}_{n}\lnot\#\mathcal{H}$ and, on the other hand, $\mathop{\mathrm{L}i}\left\{\mathcal{F}_{n}\vee\mathcal{G}_{k}:k\in\Delta n\right\}\geq\mathop{\mathrm{L}i}\left\{\mathcal{G}_{k}:k\in\Delta n\right\}\geq\mathcal{H}$, which yields a contradiction. The same argument shows that $\Delta^{-1}k$ is finite for each $k.$ It follows that

[TABLE]

Suppose that $A:=\left\{n:\Delta n=\varnothing\right\}$ is infinite. Then $\mathop{\mathrm{L}i}\nolimits_{n\in A}\mathcal{F}_{n}\geq\mathcal{H}$ and $\mathcal{F}_{n}\lnot\#\mathcal{H},$ hence by Theorem 3.1, there is a transversal subset of

[TABLE]

which yields a contradiction. Alike, one shows that $\left\{k:\Delta^{-}k=\varnothing\right\}$ is finite.

Consequently, if $A\in\mathop{\mathrm{L}i}\nolimits_{\left(n,k\right)}\left(\mathcal{F}_{n}\vee\mathcal{G}_{k}\right),$ then $A\in\mathcal{F}_{n}\cap\mathcal{G}_{k}$ for a cofinite subset of $\Delta,$ hence $A\in\mathcal{F}_{n}\cap\mathcal{G}_{k}$ for almost all $n$ (and almost all $k$), which implies that $A\in\mathcal{H}$, hence (3.6).

Let $\left\{X_{n}:n<\omega\right\}$ and $\left\{Y_{k}:k<\omega\right\}$ be sequences of disjoint sets such that $X_{n}\in\mathcal{F}_{n}$ and $Y_{k}\in\mathcal{G}_{k}$ for all $n,k<\omega.$ By definition, $\bigcup\nolimits_{\left(n,k\right)\in\Delta}\left(X_{n}\cap Y_{k}\right)\in\mathcal{H}$. Suppose that there is an infinite set $N$ such that $\mathcal{F}_{n}\ngeqslant\bigcap\nolimits_{k\in\Delta n}\left(\mathcal{F}_{n}\vee\mathcal{G}_{k}\right)$ for all $n\in N,$ then there exist $U_{n}\subset X_{n}$ such that $U_{n}\in\bigcap\nolimits_{k\in\Delta n}\left(\mathcal{F}_{n}\vee\mathcal{G}_{k}\right)\setminus\mathcal{F}_{n}$ for each $n\in N.$ Therefore $U:=\bigcup\nolimits_{n\in N}U_{n}\cup\bigcup\nolimits_{n\notin N}X_{n}\in\mathcal{H}$ from (3.6), but $U\notin\mathop{\mathrm{L}i}\nolimits_{\left(n\right)}\mathcal{F}_{n},$ which is a contradiction with (3.4).

Let $\left(\mathcal{F}_{n}\right)_{n}$ and $\left(\mathcal{H}_{p}\right)_{p}$ be sequences of filters on a set $X$. Then $\left(\mathcal{H}_{p}\right)_{p}$ is called a locally finite refinement of $\left(\mathcal{F}_{n}\right)_{n}$ if there exists a map $f:\omega\rightarrow\omega$ with finite fibers such that

[TABLE]

for almost all $n.$ A locally finite refinement is called a locally finite partition if $\left(\mathcal{H}_{p}\right)_{p}$ is disjoint.

Corollary 3.7**.**

If two totally disjoint sequences of filters have the same contour, then this is also the contour of a their common locally finite refinement.

Actually, a least common refinement of totally disjoint sequences $\left(\mathcal{F}_{n}\right)_{n}$ and $\left(\mathcal{G}_{k}\right)_{k}$ (for which (3.5) is finite-to-finite with cofinite domain and range) is given by

[TABLE]

In other words, although the contour of a sequence of filters does not determine the sequence, but it does the class of its all locally finite partitions. Therefore, two sequences of filters with the same contour are intimately related: there exists a third sequence such that each of the two consists of finite infima of the third.

4. The Stone space interpretations

Given a filter $\mathcal{H}$ on a set $X,$ we denote by $\beta\mathcal{H}$ the set of all ultrafilters finer than $\mathcal{H}$. The so defined map $\beta$ is an order isomorphism between the complete lattice of (possibly degenerate) filters on $X$ and of the complete lattice of closed sets in the Stone topology on $\beta X$ (the set of all ultrafilters on $X$). In particular,

[TABLE]

Therefore, the image by $\beta$ of the contour of $\left(\mathcal{F}_{n}\right)_{n}$ is equal to the *upper limit *of $\left(\beta\mathcal{F}_{n}\right)_{n}$ * *(333If $\tau$ is a topology, then the upper $\tau$-limit of a sequence $\left(A_{n}\right)_{n}$ of $\tau$-closed sets is defined by

$\mathop{\mathrm{L}s}\nolimits_{\left(n\right)}^{\tau}A_{n}:=\bigcap_{n<\omega}\mathop{\mathrm{c}l}\nolimits_{\tau}(\bigcup_{k>n}A_{k}).$

It is commonly called the Kuratowski-Painlevé upper limit. However, Kuratowski defines upper and lower limits in 1928, citing a paper of Painlevé, in which in fact the concept does not appear. In 1912 a Painlevé’s pupil, Zoretti, testifies that Painlevé used the notion in 1902. But the upper limit appears in the version 1902-1903 of Formulaire Mathématique of Peano, who certainly knew it well before, as he defined formally the lower limit in Applicazioni geometriche in 1887 (see [4]).), namely

[TABLE]

Consequently, in terms of the Stone topology, Theorem 3.1 becomes

Theorem 4.1** (Alternative).**

Let $\left\{A_{n}:n<\omega\right\}$ and $\left\{B_{k}:k<\omega\right\}$ be sequences of $\beta$-closed sets. If $\left(\mathop{\mathrm{L}s}\nolimits_{\left(n\right)}^{\beta}A_{n}\right)\cap\left(\mathop{\mathrm{L}s}\nolimits_{\left(k\right)}^{\beta}B_{k}\right)\neq\varnothing,$ then one of the following statements holds:

1. A.1.

$A_{n}\cap B_{k}\neq\varnothing$* for a transversal set of $\left(n,k\right)$,* 2. A.2.

$A\cap B_{k}\neq\varnothing$* for infinitely many $k,$* 3. A.3.

$A_{n}\cap B\neq\varnothing$* for infinitely many $n.$*

This means that if two upper limits in the Stone topology have non-empty intersection, then at least infinitely many terms of one sequence meet either another upper limit, or infinitely many terms of another sequence. Therefore, the behavior of upper limits with respect to the Stone topology is rather peculiar. In particular, a dual of Theorem 3.6 reads as follows (we say that a sequence $\left(H_{n}\right)_{n}$ of closed $\beta$-closed sets is totally disjoint if there is a sequence $\left(U_{n}\right)_{n}$ of disjoint $\beta$-open sets (444As each $\beta$-closed set is $\beta$-compact, we can replace in this statement open by clopen.) such that $H_{n}\subset U_{n}$ for each $n<\omega$).

Theorem 4.2**.**

If $\left(A_{n}\right)_{n}$ and $\left(B_{k}\right)_{k}$ are totally disjoint sequences of $\beta$-closed sets and $H:=\mathop{\mathrm{L}s}\nolimits_{\left(n\right)}^{\beta}A_{n}=\mathop{\mathrm{L}s}\nolimits_{\left(k\right)}^{\beta}B_{k}$, then

[TABLE]

where $\left(n,k\right)$ denotes the cofinite filter on $\Gamma:=\left\{\left(n,k\right):A_{n}\cap B_{k}\neq\varnothing\right\}.$ Moreover, $A_{n}=\bigcup_{k\in\Gamma n}\left(A_{n}\cap B_{k}\right)$ and $B_{k}=\bigcup_{n\in\Gamma^{-1}k}\left(A_{n}\cap B_{k}\right).$

In particular (555We are grateful to professor T. Nogura for having pointed out a mistake in a preliminary version of this paper.),

Corollary 4.1**.**

If $A$ and $B$ are countable sets of totally disjoint ultrafilters such that

[TABLE]

then $A$ and $B$ are almost equal.

The condition of being totally disjoint cannot be dropped.

Example 4.3**.**

Let $\left(p_{n}\right)_{n}$ be a totally disjoint sequence of ultrafilters. Hence $\partial_{\beta}\left\{p_{m}:m<\omega\right\}\cap\left\{p_{n}:n<\omega\right\}=\varnothing.$ Let $\left(q_{k}\right)_{k}$ be a sequence of distinct ultrafilters such that $q_{2k}=p_{k}$ and $q_{2k+1}\in\partial_{\beta}\left\{p_{m}:m<\omega\right\}$ for each $k<\omega.$ Then $\partial_{\beta}\left\{q_{2k+1}:k<\omega\right\}\subset\partial_{\beta}\left\{p_{m}:m<\omega\right\}$

[TABLE]

but $\left\{q_{k}:k<\omega\right\}\setminus\left\{p_{m}:m<\omega\right\}$ is infinite.

5. Sequential cascades and contours

A cascade $\left(V,\sqsubset\right)$ is a tree with a least element $\varnothing_{V},$ well-founded for the inverse order (666That is, the set $\max A,$ of maximal elements of each non empty subset $A$ of $T$, is nonempty). A cascade is called sequential if for every $v\in V\setminus\max V$, the set $V^{+}\left(v\right)$ of immediate successors of $v$ is countably infinite (777See S. Dolecki, F. Mynard [5], S. Dolecki [3], S. Dolecki, A. Starosolski, S. Watson [7].).

Each cascade is order-isomorphic to a full, closed downwards subtree of the sequential tree $\Sigma$ (888That is, the set of finite sequences of natural numbers (the empty set is denoted by $\varnothing$). If $s$ and $t$ are elements of $\Sigma$, then the concatenation of $s$ and $t$ is denoted by $s\frown t$. The abbreviation $(s,n)$ for $s\frown(n)$ (where $s\in\Sigma$ and $n<\omega$) is a useful abuse of notation. By definition, $s<t$ if there is a non-empty finite sequence $r$ such that $t=s\frown r$. With so defined partial order $\Sigma$ becomes a tree.)(999A subset $T$ of a partially ordered set $\left(\Sigma,\sqsubset\right)$ is closed downwards if $v\sqsubset s\in T$ implies $v\in T$.)(101010A subset $T$ of $\Sigma$ is called full if $T\cap\Sigma^{+}(s)\neq\varnothing$ implies that $\Sigma^{+}(s)\subset T$ for every $s\in\Sigma$.). The standard convergence $\sigma$ on $\Sigma$ (111111By a convergence on $X$ we understand a relation $x\in\lim\mathcal{F}$ between filters $\mathcal{F}$ on $X$ and elements $x$ of $X$ such that $\mathcal{F}\leq\mathcal{G}$ implies $\lim\mathcal{F}\subset\lim\mathcal{G}$, and $x\in\lim\{x\}^{\uparrow}$ where $\{x\}^{\uparrow}$ stands for the principal ultrafilter of $x$. If $\mathcal{B}$ is a filter base, then we often abridge \lim\mathcal{B}\for the limit of the filter generated by $\mathcal{B}$.) is that for which the cofinite filter on $\Sigma^{+}\left(s\right)$ converges to s\(121212The cofinite filter $\left(A\right)_{0}$ of an infinite subset $A$ of a set $X$ is defined in [6] by

$\left(A\right)_{0}:=\left\{F\subset X:\mathop{\mathrm{c}ard}\left(A\setminus F\right)<\aleph_{0}\right\}.$

). The finest topology $\mathop{\mathrm{T}}\sigma$ compatible with $\sigma$ is called the standard topology. The standard convergence $\left.\sigma\right|_{V}$ induced on a sequential cascade $V$ and the induced standard topology fulfill

[TABLE]

Let $V$ be a sequential cascade. The elements of $\mathop{\mathrm{e}xt}V=\{\varnothing_{V}\}\cup\max V$ are called extremal and the restriction of the standard topology of $V$ to $\mathop{\mathrm{e}xt}V$ will be called an Arens topology. Each Arens topology is prime, that is, such that there is at most one non isolated element. Such spaces were applied in the study of subsequential topologies by Franklin and Rajagopalan in [9]. The rank $r_{V}(v)$ of $v\in V$ is defined inductively to be [math] if $v\in\max V$, and otherwise the least ordinal greater than the ranks of the successors of $v$. The rank $r(V)$ of a sequential cascade $V$ is by definition the rank of $\varnothing_{V}$. It is always a countable ordinal.

The level $l_{V}(\varnothing)=0$ and for $v\neq\varnothing_{V}$ the *level *of $v$ is defined by

[TABLE]

As $\left\{s\in V:s\sqsubset v\right\}$ is well ordered, the level $l_{V}(v)$ is finite for every $v\in V.$ The set

[TABLE]

is called the spectrum of $v$ in $V.$ It is helpful to have in mind that each element of a cascade has finite spectrum.

The (sequential) contour $\int V$ of a sequential cascade $V$ is defined, by induction. If $r\left(V\right)=0,$ then $\int V$ is the principal filter of $\varnothing_{V};$ otherwise,

[TABLE]

where $V\left(v\right)$ is the cascade formed by all the successors of $v\in V,$ and the exterior contour is taken over the cofinite filter $\left(V^{+}\left(\varnothing_{V}\right)\right)_{0}$ on the set $V^{+}\left(\varnothing_{V}\right),$ of the immediate successors of $\varnothing_{V}$. It turns out that $\int V$ coincides with the trace on $\max V$ of the neighborhood filter of $\varnothing_{V}$ with respect to the standard topology, that is, with the neighborhood filter of $\varnothing_{V}$ of the corresponding Arens topology [5]. The rank of a sequential contour $\mathcal{F}$ on $X$ is the least ordinal $r(V)$ of a sequential cascade $V$ such that $\mathcal{F}=\int V.$

A sequential cascade $V$ is called monotone if for each $v\in V\setminus\max V$ the set $V^{+}\left(v\right),$ of immediate successors of $v,$ admits an order of the type $\omega_{0}$ for which $r_{V}$ is non-decreasing. A sequential cascade $V$ is called asymptotically monotone if the rank function $r_{V}$ is lower semicontinuous. A sequential contour $\mathcal{F}$ is called monotone if there is a monotone (equivalently, an asymptotically monotone) sequential cascade $V$ such that $\mathcal{F}=\int V$ (131313Each asymptotically monotone sequential cascade is homeomorphic to a *monotone *sequential cascade [7][3].). Monotone sequential contours have particularly simple structure among sequential contours.

5.1. Notation

Let $\left(V,\sqsubset\right)$ be a cascade. If $I\subset\left[0,r\left(V\right)\right],$ then we write

[TABLE]

and, in particular, $V^{\alpha}=\left\{v\in V:r_{V}\left(v\right)=\alpha\right\}.$

We define the relations (141414The notation is different from that used in [18].)

[TABLE]

Of course, $V^{\downarrow}=\left(V^{\uparrow}\right)^{-},$ that is, $V^{\downarrow}$ is the inverse relation of $V^{\uparrow}$. As it is customary, the image of an element $v$ of $V$ by $V^{\uparrow}$ is $V^{\uparrow}\left(v\right),$ and $V^{\uparrow}\left[A\right]$ denotes the image of a set $A$ by $V^{\uparrow}.$ However, we shall use the following abbreviations:

[TABLE]

If $V$ is a cascade and $S\subset\max V,$ then

[TABLE]

Lemma 5.1**.**

Let $V$ be a monotone sequential cascade. If $S\#\int V,$ then there is a biggest subset $S_{\infty}$ of $S$ such that

[TABLE]

is a monotone sequential cascade.

Proof.

The condition $S\#\int V$ is equivalent to $\varnothing_{V}\in\left[S_{\#}V\right].$ As (5.1), the set $\left[S_{\#}V\right]_{1}:=\left[S_{\#}V\right]\cap V^{+}\left(\varnothing_{V}\right)$ is infinite if $\varnothing_{V}\notin\max,$ that is, if $r\left(V\right)>0.$ By induction, for every $v\in\left[S_{\#}V\right]_{n}\setminus\max V,$ the set $\left[S_{\#}V\right]\cap V^{+}\left(v\right)$ is infinite. Let

[TABLE]

Then

[TABLE]

is the required monotone sequential cascade and $S_{\infty}=\mathrm{max\,}V_{S}.$ Indeed, if $v\in\mathrm{max\,}V_{S},$ then there is $n<\omega$ such that $v\in\left[S_{\#}V\right]_{n}\cap S.$ If $x\sqsubset v,$ then $\left[S_{\#}V\right]\cap V^{+}\left(x\right)$ is infinite. If $v\in\mathrm{max\,}V\setminus\mathrm{max\,}V_{S},$ then there exists $x\sqsubset v$ such that $x\notin\left[S_{\#}V\right].$

As $V_{S}$ is a relation, we shall write $V_{S}\left[B\right]$ and $V_{S}^{-}\left[B\right]$ for the image and the preimage of $B$ by $V_{S}.$

6. Heredity of the grill relation of sequential cascades

Observe that if $V$ and $W$ are monotone sequential cascades such that $\int V\geq\int W,$ then $\int V\lnot\#\int W\left(w\right)$ for each $w\in W\setminus\left\{\varnothing_{W}\right\}.$

Theorem 6.1**.**

Let $V,W$ be monotone sequential cascades with $\max V\subset X$ and $\max W\subset Y$ and let $\varphi:X\rightarrow Z,\psi:Y\rightarrow Z.$ If

[TABLE]

then there exist $R\subset X$ and $S\subset Y$ such that for each $v\in V_{R}$ there exists $w\in W_{S}$ for which

[TABLE]

and for each $w\in W_{S}$ there exists $v\in V_{R}$ such that (6.2) holds.

Proof.

In the proof, $V,W,V_{n},W_{k}$ and so on, stand for monotone sequential cascades. Order $\mathop{\mathrm{O}rd}\times\mathop{\mathrm{O}rd}$ as follows:

[TABLE]

if either $\max\left(\alpha,\beta\right)<\max\left(\delta,\gamma\right)$ or $\max\left(\alpha,\beta\right)=\max\left(\delta,\gamma\right)$ and $\min\left(\alpha,\beta\right)<\min\left(\delta,\gamma\right).$ The relation $\prec$ is well-founded (that is, each non-empty subset of $\mathop{\mathrm{O}rd}\times\mathop{\mathrm{O}rd}$ has an $\prec$-minimal element, and $\left\{\left(\alpha,\beta\right):\left(\alpha,\beta\right)\prec\left(\delta,\gamma\right)\right\}$ is a set) and we use it below for induction [12, Theorem 25’].

If $\min\{r(V),r(W)\}=0$, say $r(W)=0,$ then $\psi(\int W)$ is a principal ultrafilter, say generated by $z$. Since (6.1), $\left\{z\right\}\#\varphi\left(\int V\right)$ and so $\varphi^{-1}\left(z\right)\#\int V.$ It suffices to take $R=\max V_{\varphi^{-1}\left(z\right)}.$

So let $r\left(V\right)=\alpha>0$ and $r\left(W\right)=\beta>0$ and assume that the claim is true for $V^{\prime}$ and $W^{\prime}$ such that $\left(r\left(V^{\prime}\right),r\left(W^{\prime}\right)\right)\prec\left(\alpha,\beta\right).$ Then $\varphi\left(\int V\right)=\mathop{\mathrm{L}i}\nolimits_{\left(n\right)}\varphi(\int V_{n})$ with $r\left(V_{n}\right)<\alpha$ and $\psi\left(\int W\right)=\int_{\left(k\right)}\psi(\int W_{k})$ with $r\left(W_{k}\right)<\beta$ for each $n,k<\omega.$

If (6.1) holds, then, by Theorem 3.1, one of the following claims is true:

1. (1)

there is a transversal $A$ such that $\varphi$($\int V_{n})\#\psi(\int W_{k})$ for each $\left(n,k\right)\in A,$ 2. (2)

$\varphi$($\int V_{n})\#\psi(\int W)$ for infinitely many $n,$ 3. (3)

$\psi(\int W_{k})\#\varphi(\int V)$ for infinitely many $k.$

If (1) then $r\left(V_{n}\right)<\alpha$ and $r\left(W_{k}\right)<\beta,$ hence $\left(r\left(V_{n}\right),r\left(W_{k}\right)\right)\prec\left(\alpha,\beta\right)$. By the inductive assumption, for each $\left(n,k\right)\in A$ there exist $R_{n,k},S_{n,k}$ such that for each $v\in V_{R_{n,k}}$ there is $w\in W_{S_{n,k}}$ and for each $w\in W_{S_{n,k}}$ there is $v\in V_{R_{n,k}}$ such that

[TABLE]

Then

[TABLE]

satisfy the claim.

If (2), let $B$ be the set of $n$ for which $\varphi(\int V_{n})\#\psi(\int W).$ Then $\left(r\left(V_{n}\right),r\left(W\right)\right)\prec\left(\alpha,\beta\right),$ hence by the inductive assumption, for each $n\in B$ there exist infinite sets $R_{n},S_{n}$ such that such that for each $v\in\left(V_{n}\right)_{R_{n}}$ there is $w\in W_{S_{n}}$ and for each $w\in W_{S_{n}}$ there is so $v\in\left(V_{n}\right)_{R_{n}}$ that

[TABLE]

Then

[TABLE]

satisfies the claim. If (3) then argue mutatis mutandis as for (2).

It should be noted that Theorem 6.1 is strictly stronger than its unilateral variants like that in Corollary 6.3. In fact,

Example 6.2**.**

Consider two copies $V,W$ of $\left\{s\in\Sigma:l\left(s\right)\leq 2\right\},$ where $\Sigma$ stands for the sequential tree (66footnotemark: 6). Let $\varphi:\max V\rightarrow\max W$ be a bijection given by

[TABLE]

For $R=\max V,S=\max W,$ we have $V_{R}=V$ and $W_{S}=W.$ Then $\int V\#\int W,$ $\int W\left(\left(n\right)\right)=\int W\left(\left(n+1\right)\right)$ and $\int W\left(s\right)=\int W\left(\varphi^{-1}\left(s\right)\right)$ for $s\in\max W.$ However, $\int V\left(\left(0\right)\right)$ does not mesh $\int W\left(s\right)$ for each $l\left(s\right)\leq 2.$

Corollary 6.3**.**

If $V,W$ are monotone sequential cascades such that

[TABLE]

then there exists R\such that for each $v\in V_{R}$ there exists $w\in W$ for which

[TABLE]

The heredity property of the meshing relation, described in Corollary 6.3, can be sharpened when $\varphi(\int V)\#\psi(\int W)$ is strengthened to $\varphi(\int V)\geq\psi(\int W).$

Theorem 6.4**.**

Let $V,W$ be monotone sequential cascades and let $\varphi:\max V\rightarrow\max W$ be such that $\varphi(\int V)\geq\int W.$ Then there exists an (infinite) set $T$ such that for each $v\in V_{T}$ there exists a unique $w\left(v\right)$ for which

[TABLE]

Proof.

By Corollary 6.3, there exists a set $R$ such that for every $v\in V_{R}$ there exists $w\in W$ such that (6.3) holds with $\psi$ being the identity. To simplify, let $V:=V_{R}$ and then the condition becomes: for each $v\in V$ there is $w\in W$ such that $\varphi(\int V\left(v\right))\#\int W\left(w\right).$ In particular, if $v\in V^{1}$ then we pick any $\sqsubset_{W}$-maximal $w\left(v\right)$ with this property. Notice that $r_{W}\left(w\left(v\right)\right)=1$ if $W\left(w\left(v\right)\right)$ is free and $r_{W}\left(w\left(v\right)\right)=0$ otherwise.

We shall construct a decreasing sequence of subsets $\left\{T_{\alpha}:\alpha\leq r\left(V\right)\right\}$ of $\max V$ such that for each $\alpha\leq r\left(V\right)$ and for every $\beta<\alpha,$

[TABLE]

Let $T_{0}:=\max V$ and let

[TABLE]

Clearly, $T_{1}$ fulfills (6.5-6.7) (here $\alpha=1,\beta=0)$. Proceeding by induction on $1<\alpha\leq r\left(V\right)$, we assume that $T_{\beta}$ has been defined for each $\beta<\alpha$ and the claim holds for $\left\{T_{\beta}:\beta<\alpha\right\}.$ Define

[TABLE]

We claim (151515In particular, by (6.5), $V_{S_{\alpha}}$ is a full subcascade of $V$ such that $V^{[\alpha,r\left(V\right)]}=V_{S_{\alpha}}^{[\alpha,r\left(V\right)]}.$) that for each $v\in V^{\alpha},$ the set $M\left(v\right)$ consisting of such $w\in W$ that there exists an infinite subset $B$ of $V_{T_{<\alpha}}^{+}\left(v\right)$ for which

[TABLE]

is not empty. By (6.5), $V_{T_{\alpha}}$ is a full subcascade of $V$ such that $V^{[\alpha,r\left(V\right)]}=V_{T_{\alpha}}^{[\alpha,r\left(V\right)]},$ hence $\varnothing_{W}\in M\left(v\right)$ with $B=V_{T_{<\alpha}}^{+}\left(v\right).$ As branches of a cascade are finite, there exist $\sqsubset_{W}$-maximal elements of $M\left(v\right).$ Let

[TABLE]

be any of them, and let $B\left(v\right)$ be any infinite subset of $V_{T_{<\alpha}}^{+}\left(v\right)$ corresponding to $w\left(v\right)$ in (6.8).

Let

[TABLE]

Mind that the set $V\left[V^{(r\left(V\right),\alpha]}\setminus V^{\alpha}\right]$ consists of those $x$ for which

[TABLE]

To see that (6.5) holds, let $v\in V^{\alpha}$. Then $B\left(v\right)\subset B\left[V^{\alpha}\right],$ hence $r\left(b\right)<\alpha$ for each $b\in B\left(v\right)$, so that, by inductive (6.5), $T_{r\left(b\right)}\#\int V_{T_{r\left(b\right)}}\left(b\right),$ and by inductive (6.6), $V\left(b\right)\cap T_{<\alpha}=V\left(b\right)\cap T_{r\left(b\right)}.$ As $B\left(v\right)$ is infinite, $V\left(v\right)\cap T_{\alpha}=\bigcup_{b\in B\left(v\right)}\left(V\left(b\right)\cap T_{<\alpha}\right),$ then $T_{\alpha}\#\int V_{T_{\alpha}}\left(v\right)$ (161616If $H\#\mathcal{A}_{n}$ for infinitely many $n$ then $H\#\int_{\left(n\right)}\mathcal{A}_{n}.$).

To see (6.6), let $x\in V^{\beta}$ such that $\left\{u\in V^{(\beta,\alpha])}:u\sqsubset x\right\}=\varnothing.$ By the second part of (6.10), $T_{<\alpha}\cap V\left[V^{(\alpha,r\left(V\right)]}\setminus V^{\alpha}\right]\subset T_{\alpha}$ so that

[TABLE]

To see (6.7), let $x\in V_{T_{\alpha}}^{\beta}.$ Then either

1. (1)

there exist $v\in V^{\alpha}$ and $y\in V^{+}\left(v\right)$ such that $y\sqsubseteq x,$ 2. (2)

or $x\notin V\left(V^{\alpha}\right).$

If (1) then, by inductive assumption, $V\left(x\right)\cap T_{r\left(x\right)}=V\left(x\right)\cap T_{r\left(y\right)}$ and $V\left(y\right)\cap T_{r\left(y\right)}=V\left(y\right)\cap T_{\alpha}$ by the first part of (6.10), so $V\left(x\right)\cap T_{r\left(y\right)}=V\left(x\right)\cap T_{\alpha},$ because $V\left(x\right)\subset V\left(y\right).$ If (2), then by the second part of (6.10),

[TABLE]

thus, by the inductive assumption (6.7), we are done.

Let $T:=T_{r\left(V\right)}.$ The proof will be complete if we show that for each $v\in V_{T},$ the set

[TABLE]

is either empty or a singleton. Indeed, $\varphi\left(\int V\right)\geq\int W$ implies that $\varphi\left(T\right)\#\varphi\left(\int V\right)\#\int W,$ thus $\varphi\left(\int V_{T}\right)\geq\int W,$ hence by Corollary 6.3, there exist $R,S$ such that for each $v\in V_{T\cap R}$ there is a $w\in W_{S}$ such that $\varphi\left(\int V_{T\cap R}\left(v\right)\right)\#\int W_{S}\left(w\right).$

Lemma 6.5**.**

For each $v\in V_{T},$ the set (6.11) is either empty or a singleton.

Proof.

We shall see that if $w$ is an element of (6.11), then $w=w\left(v\right),$ as defined in (6.9). Suppose that on the contrary, there is a couple $\left(v,w\right)$ such that $\varphi\left(\int V_{T}\left(v\right)\right)\#W\int\left(w\right)$ and $w\neq w\left(v\right).$ Let $\left(v_{0},w_{0}\right)$ be the least (with respect to the lexicographic order of ranks) such a couple. Consider three cases.

1. (1)

$w\left(v_{0}\right)$ and $w_{0}$ are not $\sqsubset_{W}$-comparable, 2. (2)

$w_{0}\sqsubset_{W}w\left(v_{0}\right),$ 3. (3)

$w\left(v_{0}\right)\sqsubset_{W}w_{0}.$

Justification.

1. (1)

By (6.8) and (6.10), $\varphi\left(\max V_{T}\left(v_{0}\right)\right)\subset W\left(w\left(v_{0}\right)\right)$ and $W\left(w\left(v_{0}\right)\right)\cap W\left(w_{0}\right)=\varnothing,$ hence $\varphi\left(\int V_{T}\left(v_{0}\right)\right)$ does not mesh $\int W\left(w_{0}\right).$ 2. (2)

By (6.8) and (6.10), $\varphi\left(\max V_{T}\left(v_{0}\right)\right)\subset W\left(w\left(v_{0}\right)\right)$ and is $\int W\left(w\left(v_{0}\right)\right)$ is disjoint from $\int W\left(w_{0}\right).$ 3. (3)

By Alternative (Theorem 3.1) applied to for $\varphi\left(\int V_{T}\left(v_{0}\right)\right)$ and $\int W\left(w_{0}\right),$ one of the following holds:

1. (a)

$\varphi\left(\int V_{T}\left(v\right)\right)\#\int W\left(w\right)$ for infinitely many $v\in V_{T}^{+}\left(v_{0}\right)$ and infinitely many $w\in W^{+}\left(w_{0}\right),$ which contradicts the maximality of $w\left(v_{0}\right).$ Indeed, if $r\left(v\right)<r\left(v_{0}\right)$ then $\varphi\left(\int V_{T}\left(v\right)\right)\#\int W\left(w\right)$ implies $w=w\left(v\right)$, by minimality of $r\left(v_{0}\right)$. Therefore if $v\in V_{T}^{+}\left(v_{0}\right)$ then

[TABLE]

where $w\left(v\right)\in W^{+}\left(w_{0}\right),$ hence $w_{0}\in M\left(v_{0}\right)$, where $M$ was defined just before (6.8). 2. (b)

$\varphi\left(\int V_{T}\left(v\right)\right)\#\int W\left(w_{0}\right)$ for infinitely many $v\in V_{T}^{+}\left(v_{0}\right),$ which is contradicted analogously to the preceding case. 3. (c)

$\varphi\left(\int V_{T}\left(v_{0}\right)\right)\#\int W\left(w\right)$ for infinitely many $w\in W^{+}\left(w_{0}\right),$ in contradiction with the minimality of $r\left(w_{0}\right).$

7. Continuous maps between sequential cascades

Proposition 7.1**.**

Let $V,W$ be monotone sequential cascades such that $\varphi\left(\int V\right)\geq\int W$ and for each $v\in V$ there exists a unique $f\left(v\right)\in W$ such that $f\left(v\right)=\varphi\left(v\right)$ if $v\in\max V$ and

[TABLE]

Then there is a full subcascade $U$ of $V$ such that $\left.f\right|_{U}$ is continuous.

Proof.

If $r\left(V\right)=1$ then $r\left(W\right)=1.$ Then it is enough to set $U:=V_{\max W}.$ We use the induction with respect to the lexicographic order (initializing for each $\left(\alpha,1\right).$ If $r\left(W\right)=1,$ then $f\left(v\right)=\varnothing_{W}$ for each $v\in V\setminus\max V$.

Let $\left(\alpha,\beta\right)$ be the least couple, for which the claim does not hold, and let $V$ and $W$ witness this and let $f:V\rightarrow W$ fulfill the assumptions. Note that $\beta>1.$ By Alternative theorem, one of the following cases holds.

1. (1)

There exists an infinite subset $U_{0}$ of $V^{+}\left(\varnothing_{V}\right)$ such that $f\left(U_{0}\right)$ is an infinite subset of $W^{+}\left(\varnothing_{W}\right)$. By inductive assumption, for each $v\in U_{0}$ there exists a full subcascade $U\left(v\right)$ of $V\left(v\right)$ such that $\left.f\right|_{U\left(v\right)}$ is continuous. Then, for $U:=\bigcup_{v\in U_{0}}U_{v}\cup\left\{\varnothing_{V}\right\}$ the restriction $\left.f\right|_{U}$ is continuous. 2. (2)

There exists an infinite subset $U_{0}$ of $V^{+}\left(\varnothing_{V}\right)$ such that $f\left(U_{0}\right)=\varnothing_{W}.$ As above. 3. (3)

There exists an infinite subset $A$ of $W^{+}\left(\varnothing_{W}\right)$ such that $f\left(\varnothing_{V}\right)=a$ for each $a\in A,$ which yields a contradiction.

The following theorem considerably generalizes [7, Theorem 3.3], where the function $\varphi$ was supposed to be injective. The proof of [7, Theorem 3.3] was not correct.

Theorem 7.2**.**

If $\varphi:\mathop{\mathrm{e}xt}V\rightarrow\mathop{\mathrm{e}xt}W$ is continuous, then there exists a full subcascade $U$ of $V$ and a continuous map $f:U\rightarrow W$ such that $\left.f\right|_{\mathop{\mathrm{e}xt}U}=\left.\varphi\right|_{\mathop{\mathrm{e}xt}U}$.

Proof.

If $\varphi:\mathop{\mathrm{e}xt}V\rightarrow\mathop{\mathrm{e}xt}W$ is continuous, then $\varphi\left(\int V\right)\geq\int W.$ Use Theorem 6.4 and apply Proposition 7.1 to $V_{S}$ and $W.$

On the other hand, it is known that [7, Proposition 3.2]

Proposition 7.3**.**

If $U$ and $W$ are monotone sequential cascades and $f:U\rightarrow W$ is a continuous map such that $f\left(\max U\right)\subset\max W$ and $f\left(\varnothing_{U}\right)=\varnothing_{W}$, then

[TABLE]

for each $u\in U.$

As the rank is preserved by full subcascades, the two preceding fact imply [9, Lemma 8.4], that is, if $\alpha>\beta$ then there is no prime map from a space of subsequential order $\beta$ to a space of subsequential order $\alpha$ (see [7, Corollary 3.4] for terminology).

Simple examples like [7, Example 3.1], show that in general a continuous map $\varphi$ from Theorem 7.2 has no continuous extension to the whole cascade $V.$ By the way, Proposition 7.9 also implies the fact above.

Theorem 7.2 has also applications in set theory and logic for some classifications of ultrafilters on natural numbers (see [17],[19]).

Corollary 7.4**.**

If $V,W$ are monotone sequential cascades for which $\int V=\int W,$ then $r\left(V\right)=r\left(W\right).$

Proof.

Suppose that $r\left(V\right)<r\left(W\right)$ and $\int V=\int W.$ If $X\in\int V$ then there is a full subcscade $U$ of $V_{X}$ such that the map $\varphi(x):=x$ if $x\in X$ and $\varphi\left(\varnothing_{V}\right):=\varnothing_{W}$ coincides with a continuous map from $U$ to $W_{X}.$ By Proposition 7.3, this leads a contradiction, because $r\left(U\right)=r\left(V\right)<r\left(W_{X}\right)=r\left(W\right).$

This means that the rank of a contour is independent of a particular monotone sequential cascade.

Corollary 7.5**.**

If $\mathcal{F},\mathcal{G}$ are monotone sequential contours such that $r\left(\mathcal{G}\right)>r\left(\mathcal{F}\right),$ then $f\left(\mathcal{F}\right)\ngeqslant\mathcal{G}$ for each map $f$.

A subcascade $U$ of a cascade $V$ is called eventual if $\varnothing_{U}:=\varnothing_{V}$ and $U^{+}\left(v\right)$ is a cofinite subset of $V^{+}\left(v\right)$ for each $v\in U.$ Let $V,T$ be monotone sequential cascades. We say that $T$ is a *locally finite partition *of $V$ (in symbols, $T\vartriangleright V)$ if there exist an eventual subcascade $U$ of $V$ and a surjection $f:T\rightarrow V$ with finite fibers such that

[TABLE]

This notion was introduced in [18] (171717It was formulated slightly differently.).

Proposition 7.6**.**

If $T\vartriangleright V,$ then $\int T=\int V.$

Proof.

The claim is obviously true if $r\left(V\right)=1.$ Let $r\left(V\right)=\alpha>1$ and suppose that the claim is true if $r\left(V\right)=\beta$ for each $\beta<\alpha.$ Then

[TABLE]

where in all the cases above, $\mathop{\mathrm{L}i}$ are taken over the corresponding cofinite filters.

Theorem 7.7**.**

If $V,W$ are monotone sequential cascades such that $\int V=\int W,$ then there exists a monotone sequential cascade $T$ such that $T\vartriangleright V$ and $T\vartriangleright W.$

Proof.

By Corollary 7.5, $r\left(V\right)=r\left(W\right).$ We induce on the common rank of $V$ and $W.$ If $r\left(V\right)=r\left(W\right)=1,$ let $S:=\max V\cap\max W,$ and $T:=V_{S}=W_{S}.$ Then $\int T=\int V=\int W,$ because $T$ is cofinite in $\max V$ and in $\max W.$

Let $r\left(V\right)=r\left(W\right)=\alpha>1$, and suppose that the claim is true for each $\beta<\alpha.$ Represent $V^{+}\left(\varnothing_{V}\right)=\left\{v_{n}:n<\omega\right\}$ and $W^{+}\left(\varnothing_{W}\right)=\left\{w_{k}:k<\omega\right\}.$ By Theorem 3.6, the relation

[TABLE]

is finite-to finite, and its domain and range are cofinite. Moreover,

[TABLE]

where $\left(n,k\right)$ stands for the cofinite filter on $R.$ We start defining a cascade $T,$ and maps $f_{V}:T\rightarrow V$ and $f_{W}:T\rightarrow W,$ by letting $T^{+}\left(\varnothing_{T}\right):=\left\{t_{n,k}:\left(n,k\right)\in R\right\}$ and let

[TABLE]

be defined by $f_{V}\left(t_{n,k}\right):=v_{n}$ and $f_{W}\left(t_{n,k}\right):=w_{k}.$ By inductive assumption, for each $\left(n,k\right)\in R,$ there exists a cascade $T_{n,k}$ such that maps $\varphi_{n,k,V}:T_{n,k}\rightarrow V\left(v_{n}\right)$ and $\varphi_{n,k,W}:T_{n,k}\rightarrow W\left(w_{k}\right)$ fulfilling the conditions, mutatis mutandis, (7.2)-(7.4). We accomplish defining a cascade $T$ for $t\sqsupset_{T}\left(n,k\right)$ by adjoining to each $(n,k)$ the cascade $T_{n,k},$ and maps $f_{V}:T\rightarrow V$ and $f_{W}:T\rightarrow W,$ by gluing $\varphi_{n,k,V}$ to $f_{V}$ and $\varphi_{n,k,W}$ to $f_{W}$ (see [5] for details).

A full subcascade $U$ of $V,$ to which the map $\varphi$ can be extended in Theorem 7.2, is not unique in general. However, we cannot choose a priori an ultrafilter $\mathcal{Z}$ finer than $\int V$ so that $\max U\in\mathcal{Z}$.

Example 7.8**.**

Consider a monotone sequential cascade $V$ of rank $2.$ Let $V^{+}\left(\varnothing_{V}\right)=\left\{v_{n}:n<\omega\right\}$. Denote $X_{n}:=\max V\left(v_{n}\right)$ and, for each $0<n<\omega,$ split $X_{n}$ into $n$ disjoint infinite sets $X_{n}=X_{n,1}\cup X_{n,2}\cup\ldots\cup X_{n,n.}$ Define a sequential monotone cascade $W$ so that $\left\{X_{n,k}:0<k\leq n,n<\omega\right\}$ coincide with $\left\{\max W\left(w\right):r_{W}\left(w\right)=1\right\}$, and consider the identity $i:\mathop{\mathrm{e}xt}V\rightarrow\mathop{\mathrm{e}xt}W.$

Let $\mathbb{U}$ the set of $U,$ for which $i$ can be continuously extended to a map from $U$ to $W.$ By Theorem 7.2, $\mathbb{U}\neq\varnothing$ and $U\cap\left\{v_{n}:n<\omega\right\}$ is infinite for each $U\in\mathbb{U}$.

We shall construct an ultrafilter $\mathcal{Z}$ such that $\mathcal{Z}\geq\int V=\int W$ and $\max U\notin\mathcal{Z}$ for each $U\in\mathbb{U}$. Let $\mathcal{A}$ be the set of $A\subset\bigcup_{n<\omega}X_{n}$ such that $\mathop{\mathrm{c}ard}\left\{k:X_{n,k}\setminus A\neq\varnothing\right\}\leq 1$ for each $n<\omega.$ Then $\mathcal{A}\#\int V,$ because if $A_{1},\ldots,A_{p}\in\mathcal{A}$, then $A_{1}\cap\ldots\cap A_{p}\cap X_{n}\neq\varnothing$ for $n>p.$ Let $\mathcal{Z}$ be an ultrafilter finer than $\mathcal{A}\vee\int V.$ If $U\in\mathbb{U},\max U\in\mathcal{Z}$ and $f:U\rightarrow W$ is a continuous extension of $i:\mathop{\mathrm{e}xt}V\rightarrow\mathop{\mathrm{e}xt}W,$ then $\max U\in\mathcal{A}^{\#}$ and $\max U\cap X_{n}\neq\varnothing$ for infinitely many $n.$ If $1\leq k\leq n$ is such that $f\left(v_{n}\right)\in X_{n,k}$ then $X_{n}\supset X_{n,k}\supset f^{-1}\left(\left(V^{+}\left(f\left(v_{n}\right)\right)\right)_{0}\right)$ is not cofinite in $X_{n}$ for $n>1,$ hence $f$ is not continuous at $v_{n}.$

This example extends easily to

Proposition 7.9**.**

For every monotone sequential cascade $V$ of rank greater than $1,$ there exist a monotone sequential cascade $W$ and an ultrafilter $\mathcal{Z}\geq\int V=\int W$ such that if $U$ is a full subcascade of $V,$ for which there exists a continuous extension $f:U\rightarrow W$ of the identity $i:\max V\rightarrow\max W,$ then $\max U\notin\mathcal{Z}$.

Theorem 7.2 can be interpreted in terms of subsequential spaces. A topological space is called prime if at most one point is not isolated. A map $\varphi$ from a prime space $X$ to a prime space is called non-trivial if it maps isolated points into isolated points, and the non-isolated point of $X$ onto the non-isolated point of $Y.$ A subset $V$ of a prime space $X$ with a non-isolated point $o_{X}$ is called full if $o_{X}\in\mathop{\mathrm{c}l}\nolimits_{X}V\setminus V.$

Proposition 7.10**.**

Let $X$ and $Y$ be prime subsequential topological spaces and $\varphi:X\rightarrow Y$ is a continuous non-trivial map. Then there exist sequential spaces $\widetilde{V}$ and $\widetilde{Y}$, a full subset $V$ of $X$ and a continuous map $f:\widetilde{V}\rightarrow\widetilde{Y}$ such that $V$ is a dense subset of $\widetilde{V},Y$ is a dense subset of $\widetilde{Y}$ and $\left.f\right|_{V}=\left.\varphi\right|_{V}.$

8. Appendix: Contour operation

We considered the contour $\mathop{\mathrm{L}i}\nolimits_{\left(n\right)}\mathcal{F}_{n}$ of a sequence $\left(\mathcal{F}_{n}\right)_{n}$ of filters. More generally, if $\mathcal{B}\left(y\right)\subset 2^{X}$ for each $y\in Y,$ then the contour of $\left\{\mathcal{B}\left(y\right):y\in Y\right\}$ along $\mathcal{A}\subset 2^{Y}$ is defined by

[TABLE]

that is, the set-theoretic lower limit of subsets of $2^{X}$ along a family $\mathcal{A}$. In this framework, the contour of a sequence is that of $\left\{\mathcal{F}_{n}:n<\omega\right\}$ along the cofinite filter $\left(n\right),$ on the natural numbers (181818In some previous papers ([5],[7],[3]), it was denoted by $\int_{\left(n\right)}\mathcal{F}_{n},$ that is,

$\int_{\left(n\right)}\mathcal{F}_{n}:=\mathop{\mathrm{L}i}\nolimits_{\left(n\right)}\mathcal{F}_{n}.$

). This notion was introduced by G. H. Greco in [11, p. 158] for his theory of limitoids with important applications to the theory of $\Gamma$-functionals.

In the case of filters, the operation (8.1) was introduced by H. J. Kowalsky in [13] (191919Kowalsky inverted the order in the lattice of filters, so that $\bigcap$ and $\bigcup$ should be interchanged when translated to the usual order (for which ultrafilters are maximal filters).), where it was used for a characterization of diagonality of convergence spaces [14]. In [2], C. H. Cook and H. R. Fisher referred to Kowalsky’s operation (202020Defined as the set-theoretic upper limit (for the inverse order).), under the name of compression operator, to characterize regularity of convergence spaces. As mentioned in Section 1, the contour can be employed, in two complementary ways, for the characterization of both diagonality and regularity [6, Chapter V].

Z. Frolík introduced (8.1) in [10] in the case of ultrafilters, calling it a sum of ultrafilters, and used it to prove in ZFC his famous theorem on non-homogeneity of the Stone topology, showing that there are $2^{\mathfrak{c}}$ types of ultrafilters.

In [6, p. 53] S. Dolecki and F. Mynard presented a more abstract vision of the contour operation, namely, for each family $\mathcal{H}$ of subsets of the set of families of subsets of a given set $X$ (212121Specifically, in order to avoid an accumulation of exponents on five levels, they denote $\S X:=2^{\left(2^{X}\right)},$ so that $\mathcal{H}\in\S\left(\S X\right)$ and thus $\mathcal{H}^{\ast}\in\S X.$), the contour $\mathcal{H}^{\ast}$ is defined by

[TABLE]

The definition (8.1) can be easily seen to be a special case of the notion above.

Considering the set-theoretic meaning of the notion, the term lower limit would be probably most precise to identify the object. However, we opted for a shorter term contour, intended to suggest a limiting effect of a family $\mathcal{A}$ on the family of families $\left\{\mathcal{B}\left(y\right):y\in Y\right\}.$ The term sum is most frequently used by general topologists.