On condensations onto $\sigma$-compact spaces

In this paper, we prove the following result. Let $X$ be a complete metric space of weight $w(X)$ and $H\subseteq X$ be a set such that $w(X)<|H|<c$. Then there is no continuous bijection of the subspace$X\setminus H$ onto a $\sigma$-compact space. As a result, there is no continuous bijection of the subspace $X\setminus H$ onto a Polish space. Thus, it has been proved that metric compact spaces are not $a_\tau$-spaces for any uncountable cardinal number $\tau$. This result answers the question asked by E.G. Pytkeev in his coauthored work “On the properties of subclasses of weakly dyadic compact sets” to be published in the Siberian Mathematical Journal.