Lemma

Let T be a completely regular space, K a compact subset of T, and U an open subset of T containing K. a) There exists an open subset U' of T such that $K \subset U' \subset \overline{U'} \subset U$. b) Let f be a continuous function defined on K with values in an interval I of $\mathbf{R}$ (resp. in $\mathbf{C}$). There exists a bounded continuous function $f'$ on T, with values in I (resp. in $\mathbf{C}$), that extends f and is zero on $T - U$.

It suffices to treat the case that T is a subspace of a compact space X. Let V be an open subset of X such that $V \cap T = U$; denote by $V'$ an open set in X containing K such that $\overline{V'} \subset V$, by g a continuous function on X with values in I (resp. in $\mathbf{C}$) extending f and zero on $X - V$ (GT, IX, §4, No. 1, Prop. 1). The condition a) is satisfied by taking $U' = V' \cap T$, and b) by taking $f'$ to be the restriction of g to T.