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Zbl 1199.54099
Larson, Suzanne
Functions that map cozerosets to cozerosets. (English)
[J] Commentat. Math. Univ. Carol. 48, No. 3, 507-521 (2007). ISSN 0010-2628

Several examples of properties of topological spaces are investigated which are not preserved by continuous open mappings. Subclasses of continuous mappings that map cozero sets to cozero sets are studied in connection with preservation of such properties. The main result in this direction says that the Stone extension of a continuous $z$-open function is open. This generalizes a result by {\it T. Isiwata} [Pac. J. Math. 20, 455--480 (1967; Zbl 0149.40501); correction 23, 630--631 (1967)]. Here, $f\: X\to Y$ is $z$-open if, for every pair $Z\subset H$, $Z$ a zero, and $H$ a cozero set, we have $\operatorname {cl}_Y(f(Z))\subset \operatorname {int}_Y(f(H))$. As a corollary, the preservation of $F$-spaces, finite rank spaces, and several other notions are discussed.
[Petr Holický (Praha)]

MSC 2000:: *54C10 Special maps on topological spaces
54G05 Extremally disconnected spaces, etc.

Keywords: open function; cozeroset preserving function; $z$-open function; $F$-space; $SV$ space; finite rank

Citations: Zbl 0149.40501

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