On Borsuk's non-retract theorem

Abstract

Let $X$ and $Y$ be the Hausdorff topological spaces and let $A$ be both an $\fs$ and $\gd$ subset of $X$. Let also $f\cn A\to Y$ be a function for which the inverse image of every open subset $U\subset Y$ is $\fs$ in $X$. We will prove that $f$ can be linearly extended to a function defined on whole $X$. An analogous result is proved for the first class function defined on an analogous subset of $\mR$. We give also an answer when the extension map is (with a supremum norm) an isometry.

In the second part of the paper we deal with the classical Borsuk's non-retract theorem. It says that a unit sphere in $\mR^n$ is not a continuous retract of the unit closed ball. We will show that such a unit sphere is a piecewise continuous retract of the unit closed ball.