Functions that map cozerosets to cozerosets

Suzanne Larson

Commentationes Mathematicae Universitatis Carolinae (2007)

  • Volume: 48, Issue: 3, page 507-521
  • ISSN: 0010-2628

Abstract

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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 ⊆ Z ⊆ V , there is a zeroset Z ' of Y such that f ( U ) ⊆ Z ' ⊆ 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.

How to cite

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Larson, Suzanne. "Functions that map cozerosets to cozerosets." Commentationes Mathematicae Universitatis Carolinae 48.3 (2007): 507-521. <http://eudml.org/doc/250239>.

@article{Larson2007,
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 $\operatorname\{cl\}_Y(f(Z))$ 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 $Z^\{\prime \}$ of $Y$ such that $f(U)\subseteq Z^\{\prime \}\subseteq f(V)$. 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.},
author = {Larson, Suzanne},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {open function; cozeroset preserving function; z-open function; F-space; SV space; finite rank; open function; cozeroset preserving function; -open function; -space; space; finite rank},
language = {eng},
number = {3},
pages = {507-521},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Functions that map cozerosets to cozerosets},
url = {http://eudml.org/doc/250239},
volume = {48},
year = {2007},
}

TY - JOUR
AU - Larson, Suzanne
TI - Functions that map cozerosets to cozerosets
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2007
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 48
IS - 3
SP - 507
EP - 521
AB - 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 $\operatorname{cl}_Y(f(Z))$ 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 $Z^{\prime }$ of $Y$ such that $f(U)\subseteq Z^{\prime }\subseteq f(V)$. 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.
LA - eng
KW - open function; cozeroset preserving function; z-open function; F-space; SV space; finite rank; open function; cozeroset preserving function; -open function; -space; space; finite rank
UR - http://eudml.org/doc/250239
ER -

References

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