Interpolation sets in spaces of continuous metric-valued functions

Let $X$ and $M$ be a topological space and metric space, respectively. If $C(X,M)$ denotes the set of all continuous functions from X to M, we say that a subset $Y$ of $X$ is an \emph{$M$-interpolation set} if given any function $g\in M^Y$ with relatively compact range in $M$, there exists a map $f\in C(X,M)$suchthat$f_{|Y}=g$. In this paper, motivated by a result of Bourgain in \cite{Bourgain1977}, we introduce a property, stronger than the mere \emph{non equicontinuity} of a family of continuous functions, that isolates a crucial fact for the existence of interpolation sets in fairly general settings. As a consequence, we establish the existence of $I_0$ sets in every nonprecompact subset of a abelian locally $k_{\omega }$-groups. This implies that abelian locally $k_{\omega }$-groups strongly respects compactness.