Continuous extension of functions from countable sets

Abstract

We give a characterization of countable discrete subspace $A$ of a topological space $X$ such that there exists a (linear) continuous mapping $\phi :C_p^{\ast }(A)\to C_p(X)$ with $\phi (y){|}_A=y$ for every $y\in C_p^{\ast }(A)$. Using this characterization we answer two questions of A.~Arhangel'skii. Moreover, we introduce the notion of well-covered subset of a topological space and prove that for well-covered functionally closed subset $A$ of a topological space $X$ there exists a linear continuous mapping $\phi :C_p(A)\to C_p(X)$ with $\phi (y){|}_A=y$ for every $y\in C_p(A)$.