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

## TL;DR

This paper investigates the limitations on the size of continuous images of spaces with small cellularity, showing that such images can be constructed with controlled weight, depending on the cellularity of the original space.

## Contribution

It provides new bounds on the weight of nowhere constant and pseudo-open continuous images of crowded Tychonov spaces based on cellularity, introducing the use of the hat cellularity function.

## Key findings

- Any crowded Tychonov space has a nowhere constant image with weight at most the hat cellularity.
- Any crowded Tychonov space has a pseudo-open image with weight at most 2^{<hat cellularity}.
- Under Martin's axiom, there exists a space where all pseudo-open images have weight at least continuum.

## Abstract

We call a continuous map $f : X \to Y$ 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 $f^{-1}(y)$ of $f$ is nowhere dense in $X$. We call the continuous map $f : X \to Y$ pseudo-open if for each nowhere dense $Z \subset Y$ its inverse image $f^{-1}(Z)$ 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 $\widehat{c}(X)$, the "hat version" of cellularity, which is defined as the smallest cardinal $\kappa$ such that there is no $\kappa$-sized disjoint family of open sets in $X$. Thus, for instance, $\widehat{c}(X) = \omega_1$ means that $X$ is CCC. THEOREM A. Any crowded Tychonov space $X$ has a crowded Tychonov nowhere constant continuous image $Y$ of weight $w(Y) \le \widehat{c}(X)$. Moreover, in this statement $\le$ may be replaced with $<$ iff there are no $\widehat{c}(X)$-Suslin lines (or trees). THEOREM B. Any crowded Tychonov space $X$ has a crowded Tychonov pseudo-open continuous image $Y$ of weight $w(Y) \le 2^{<\widehat{c}(X)}$. 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 $w(Y) \ge \mathfrak{c}\,( = 2^{< \omega_1})$.