Spaces of small cellularity have nowhere constant continuous images of small weight

Dedicated to Professor Arhangel'skii, on the occasion of his 80th birthday
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Abstract

We call a continuous map f:XY nowhere constant if it is not constant on any non-empty open subset of its domain X. Clearly, this is equivalent with the assumption that every fiber f1(y) of f is nowhere dense in X. We call the continuous map f:XY pseudo-open if for each nowhere dense ZY its inverse image f1(Z) is nowhere dense in X. Clearly, if Y is crowded, i.e. has no isolated points, then f is nowhere constant.

The aim of this paper is to study the following, admittedly imprecise, question: How “small” nowhere constant, resp. pseudo-open continuous images can “large” spaces have? Our main results yield the following two precise answers to this question, explaining also our title. Both of them involve the cardinal function cˆ(X), the “hat version” of cellularity, which is defined as the smallest cardinal κ such that there is no κ-sized disjoint family of open sets in X. Thus, for instance, cˆ(X)=ω1 means that X is CCC.

THEOREM A. Any crowded Tychonov space X has a crowded Tychonov nowhere constant continuous image Y of weight w(Y)cˆ(X). Moreover, in this statement ≤ may be replaced with < iff there are no cˆ(X)-Suslin lines (or trees).

THEOREM B. Any crowded Tychonov space X has a crowded Tychonov pseudo-open continuous image Y of weight w(Y)2<cˆ(X). If Martin's axiom holds then there is a CCC crowded Tychonov space X such that for any crowded Hausdorff pseudo-open continuous image Y of X we have w(Y)c(=2<ω1).

Introduction

In this paper all spaces are assumed to be crowded Hausdorff spaces. A space is called crowded if it has no isolated points. Moreover, all maps considered are continuous maps between such spaces.

Of course, all spaces with additional properties are also assumed to be crowded. One, perhaps less widely known such property which we shall frequently assume is that of π-regularity. A (Hausdorff) space X is π-regular if for every non-empty open set U in X there is a non-empty open set V such that VU, in other words: the regular closed subsets of X form a π-network in X, see e.g. [7].

For any space X we denote by τ(X) the topology of X and put τ+(X)=τ(X){}. Similarly, RO(X) denotes the collection of all regular open sets in X, moreover RO+(X)=RO(X){}. We shall denote by CR(X) the collection of all non-empty crowded subspaces of the space X.

As is mentioned in the abstract, the aim of this paper is to examine what can be said about the weight of nowhere constant continuous images of spaces. Here is the self-explanatory definition of such maps.

Definition 1.1

A continuous map f:XY is nowhere constant (NWC) iff it is not constant on any non-empty open sets, i.e.Uτ+(X)|f[U]|>1.

The following simple proposition yields alternative definitions of this concept.

Proposition 1.1

For a continuous map f:XY the following statements are equivalent:

  • (a)

    f is NWC,

  • (b)

    f[U] is crowded for each Uτ+(X),

  • (c)

    f1(y) is nowhere dense in X, i.e. int(f1(y))= for each yY.

Proof

(a) ⇔ (b): It is immediate that f[X] is crowded if f is NWC. Next, if Uτ+(X) then the restriction fU is NWC as well.

(a) ⇔ (c) is obvious because f1(y) is closed for each yf[X]. 

This proposition leads us to two natural strengthenings of the notion of NWC maps that we shall also consider.

Definition 1.2

A continuous map f:XY is pseudo-open (PO) iff for every nowhere dense subset Z of Y its preimage f1(Z) is nowhere dense in X.

As singletons in a crowded space are nowhere dense, every PO map into a crowded space is NWC. The following observation is obvious, so we leave its proof to the reader.

Proposition 1.2

For a continuous map f:XY the following three statements are equivalent:

  • (a)

    f is PO,

  • (b)

    the preimage of any dense open subset of Y is dense (and open) in X,

  • (c)

    for each Uτ+(X) we have int(f[U]).

We recall that a continuous map f:XY is called quasi-open (QO) if for each Uτ+(X) we have int(f[U]), hence every QO map is PO. In some important cases the converse of this also holds.

Fact 1.3

If the PO map f:XY is also closed and X is π-regular, in particular if X is compact, then f is QO.

Now we give the second strengthening of the notion of NWC maps. This is based on the obvious fact that any non-empty open subspace of a crowded space is crowded.

Definition 1.4

A continuous map f:XY is crowdedness preserving (CP) iff the image of any crowded subspace of X is crowded.

We again have alternative characterizations of this notion.

Theorem 1.5

For any continuous map f:XY the following statements are equivalent:

  • (a)

    f is CP,

  • (b)

    |f[S]|>1 for each SCR(X),

  • (c)

    the preimage of every point of Y is scattered.

Proof

(a) ⇒ (b): If SCR(X) then f[S] is crowded and so infinite.

(b) ⇒ (c): If Sf1(y) for some yY, then f[S]{y}, and so S is not crowded by (b). Thus f1(y) does not contain any crowded subspaces, i.e. f1(y) is scattered.

(c) ⇒ (a): Assume (c) and that f[S] is not crowded for some non-empty SX. Then there is an open Wτ+(Y) such that f[S]W={f(x)} for some xS. Since f is continuous, there is an open neighborhood U of x such that f[U]W, hence f[US]={f(x)}. Thus US is scattered by (c), and so S is not crowded. Thus (a) holds. 

Section snippets

Splitting families and splitting numbers

In this section we introduce two kinds of splitting families that will turn out to play an essential role in finding NWC or CP images of “small” weight of certain spaces.

Definition 2.1

Assume that X is a space and A,BP(X). We say that

  • (1)

    A T2-splits B (or A is a T2-splitting family for B) iffBBA0,A1A(A0A1=A0BA1B).

  • (2)

    A T3-splits B (or A is a T3-splitting family for B) iffBBA0,A1A(A0A1=A0BA1B).

Actually, in all the interesting cases for us the splitting family A will consist of open sets;

Shattering, splitting, and cellularity

The aim of this section is to present the first of our two main results, Theorem A from the abstract. The crucial step will be achieved by establishing that the cellularity number cˆ(X) is an upper bound for the splitting number os2(X) (resp. os3(X)) for all (resp. all π-regular) spaces. This, in turn, will make use of the concepts of shattering family and shattering number that we shall define below.

Definition 3.1

(i) For any space X letcel(X)={SRO+(X):S is disjoint}. (ii) Fcel(X) is called a shattering

Pseudo-open images

The first result of this section, similarly to Theorem 3.5, yields an upper bound for the minimum weight of a Tychonov PO image of a Tychonov space X in terms of cˆ(X). However, as being PO is more restrictive than being NWC, it is not surprising that the upper bound for PO images is larger than the upper bound for NWC images. As we shall see later, at least consistently, this new upper bound is sharp.

Theorem 4.1

Any Tychonov space X has a Tychonov PO image Y of weight w(Y)2<cˆ(X).

Proof

We may assume that X is

The case of ω

In this section we collected everything we could prove concerning the previously discussed topics in the case of the 0-dimensional compact space ω, the Čech - Stone remainder of ω. We think it is interesting that the values on ω of the various cardinal functions we introduced above coincide with various well-known and well-studied cardinal characteristics of the continuum, see e.g. Chapter 9 of [5]. The topological facts about ω that we shall use are well known, they can be found e.g. in [9].

An application to densely k-separable spaces

We start this section with a couple of simple definitions taken from [2].

Definition 6.1

A space X is called k-separable if it has a σ-compact dense subset. We say that X is densely k-separable if every dense subspace of X is k-separable.

It was shown in [2] that every densely k-separable compact space is actually densely separable, or equivalently, has countable π-weight. The aim of this section is to present a result on NWC images of densely k-separable spaces which provides an alternative to – the lengthily

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The preparation of this paper was supported by NKFIH grants K113047 and K129211.

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