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We call a continuous map $f : X \to Y$ nowhere constant if it is not constanton any non-empty open subset of its domain $X$. Clearly, this is equivalentwith 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 nowheredense $Z \subset Y$ 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 nowhereconstant.  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 answersto this question, explaining also our title. Both of them involve the cardinalfunction $\widehat{c}(X)$, the "hat version" of cellularity, which is definedas the smallest cardinal $\kappa$ such that there is no $\kappa$-sized disjointfamily 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 nowhereconstant continuous image $Y$ of weight $w(Y) \le \widehat{c}(X)$. Moreover, inthis 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-opencontinuous image $Y$ of weight $w(Y) \le 2^{<\widehat{c}(X)}$. If Martin'saxiom holds then there is a CCC crowded Tychonov space $X$ such that for anycrowded Hausdorff pseudo-open continuous image $Y$ of $X$ we have $w(Y) \ge\mathfrak{c}\,( = 2^{< \omega_1})$.

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