Extension of point-finite partitions of unity

Haruto Ohta¹ ; Kaori Yamazaki²

¹ Faculty of Education Shizuoka University Ohya, Shizuoka 422-8529, Japan
² Institute of Mathematics University of Tsukuba Ibaraki 305-8571, Japan

Fundamenta Mathematicae, Tome 191 (2006) no. 3, pp. 187-199

Résumé

A subspace $A$ of a topological space $X$ is said to be $P^\gamma$-embedded ($P^\gamma$(point-finite)-embedded) in $X$ if every (point-finite) partition of unity $\alpha$ on $A$ with $|\alpha|\leq\gamma$ extends to a (point-finite) partition of unity on $X$. The main results are: (Theorem A) A subspace $A$ of $X$ is $P^\gamma({\rm point-finite})$-embedded in $X$ iff it is $P^\gamma$-embedded and every countable intersection $B$ of cozero-sets in $X$ with $B\cap A=\emptyset$ can be separated from $A$ by a cozero-set in $X$. (Theorem B) The product $A\times[0,1]$ is $P^\gamma({\rm point-finite})$-embedded in $X\times[0,1]$ iff $A\times Y$ is $P^\gamma({\rm point-finite})$-embedded in $X\times Y$ for every compact Hausdorff space $Y$ with $w(Y)\leq\gamma$ iff $A$ is $P^\gamma$-embedded in $X$ and every subset $B$ of $X$ obtained from zero-sets by means of the Suslin operation, with $B\cap A=\emptyset$, can be separated from $A$ by a cozero-set in $X$. These characterizations are used to answer certain questions of Dydak. In particular, it is shown that, assuming CH, the property of $A\times[0,1]$ to be $P^\gamma({\rm point-finite})$-embedded in $X\times[0,1]$ is stronger than that of $A$ being $P^\gamma({\rm point-finite})$-embedded in $X$.

Détail

DOI : 10.4064/fm191-3-1

Keywords: subspace topological space said gamma embedded gamma point finite embedded every point finite partition unity alpha alpha leq gamma extends point finite partition unity main results theorem nbsp subspace gamma point finite embedded gamma embedded every countable intersection cozero sets cap emptyset separated cozero set nbsp theorem nbsp product times gamma point finite embedded times times gamma point finite embedded times every compact hausdorff space leq gamma gamma embedded every subset obtained zero sets means suslin operation cap emptyset separated cozero set nbsp these characterizations answer certain questions dydak particular shown assuming property times gamma point finite embedded times stronger being gamma point finite embedded nbsp