  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 $w(Y) \ge
\mathfrak{c}\,( = 2^{< \omega_1})$.
