Suzanne Larson
Functions that map cozerosets to cozerosets

Comment.Math.Univ.Carolin. 48,3 (2007) 507-521.

Abstract:A function $f$ mapping the topological space $X$ to the space $Y$ is called a {z-open} function if for every cozeroset neighborhood $H$ of a zeroset $Z$ in $X$, the image $f(H)$ is a neighborhood of $cl_Y(f(Z))$ in $Y$. We say $f$ has the {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.

Keywords: open function, cozeroset preserving function, z-open function, F-space, SV space, finite rank
AMS Subject Classification: Primary 54C30, 54C10