Extension of continuous mappings and $H_1$-retracts

TL;DR

This paper demonstrates that continuous functions defined on certain metrizable subspaces can be extended to Lebesgue class one functions on larger spaces, broadening the scope of extension theorems in topology.

Contribution

It establishes a new extension result for continuous mappings from completely metrizable subspaces to arbitrary topological spaces, ensuring the extended function is of Lebesgue class one.

Findings

Extension of continuous functions to Lebesgue class one functions

Applicable to subspaces of perfect paracompact spaces

Works for arbitrary target topological spaces

Abstract

We prove that any continuous mapping $f:E\to Y$ on a completely metrizable subspace $E$ of a perfect paracompact space $X$ can be extended to a Lebesgue class one mapping $g:X\to Y$ (i.e. for every open set $V$ in $Y$ the preimage $g^{-1}(V)$ is an $F_\sigma$-set in $X$) with values in an arbitrary topological space $Y$.