Extension of point-finite partitions of unity
Volume 191 / 2006
Abstract
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 $X$.
