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open function; cozeroset preserving function; z-open function; F-space; SV space; finite rank
A function $f$ mapping the topological space $X$ to the space $Y$ is called a {\it z-open\/} function if for every cozeroset neighborhood $H$ of a zeroset $Z$ in $X$, the image $f(H)$ is a neighborhood of $\operatorname{cl}_Y(f(Z))$ in $Y$. We say $f$ has the {\it z-separation property\/} if whenever $U$, $V$ are cozerosets and $Z$ is a zeroset of $X$ such that $U\subseteq Z\subseteq V$, there is a zeroset $Z'$ of $Y$ such that $f(U)\subseteq Z'\subseteq f(V)$. A surjective function is z-open if and only if it maps cozerosets to cozerosets and has the z-separation property. We investigate z-open functions and other functions that map cozerosets to cozerosets. We show that if $f$ is a continuous z-open function, then the Stone extension of $f$ is an open function. This is used to show several properties of topological spaces related to F-spaces are preserved under continuous z-open functions.
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