Surjective characterizations of metrizable LC∞-spaces

Abstract

In this note the following theorem is proved (theorem 1.1):

A metrizable space $Y$ is $L{C}^{\mathrm{\infty }}$ (resp. $L{C}^{\mathrm{\infty }}\mathrm{\&}{C}^{\mathrm{\infty }}$) if and only if for any paracompact $p$-space $X$ and any closed locally finite-dimensionally embedded subset $A$ of $X$, any map $f:A\to Y$ can be continously extended to a neighborhood of $A$ in $X$ (resp. to $X$).

using this theorem we give a positive answer of the following question of A.Chigogidze: Is it true that a metrizable space $Y$ is $L{C}^{\mathrm{\infty }}\mathrm{\&}{C}^{\mathrm{\infty }}$ if and only if $Y$ is an image of an absolute extensor for metrizable spaces under a $\mathrm{\infty }$-soft map?