Continuous extension of functions from countable sets

TL;DR

This paper characterizes when continuous extensions of functions from countable discrete subsets of topological spaces exist, introduces well-covered subsets, and answers related questions in topology.

Contribution

It provides a new characterization for extending functions from countable discrete subsets and introduces the concept of well-covered subsets in topology.

Findings

Characterization of countable discrete subspaces allowing continuous function extension

Introduction of well-covered subsets and their properties

Answers to two open questions by A. Arhangel'skii

Abstract

We give a characterization of countable discrete subspace $A$ of a topological space $X$ such that there exists a (linear) continuous mapping $\varphi:C_p^*(A)\to C_p(X)$ with $\varphi(y)|_A=y$ for every $y\in C_p^*(A)$. 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 $\varphi:C_p(A)\to C_p(X)$ with $\varphi(y)|_A=y$ for every $y\in C_p(A)$.