Extension of continuous mappings and $H_1$-retracts
  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 $Y$. 
Olena Karlova
math.GN