Prove: $(\forall \epsilon > 0 )(\forall x \in K)(\exists U \in O(x))(\forall f \in Z)(\forall y \in U) | f(x) - f(y) | < \epsilon$ .

Question

Let $K$ be compact space and $Z \subseteq C(K, \mathbb{R})$ compact, in metric topology $T_{d_{\infty}}$ on $C(K, \mathbb{R})$ which is induced by metrics $d_{\infty}$. Prove:

$(\forall \epsilon > 0 )(\forall x \in K)(\exists U \in O(x))(\forall f \in Z)(\forall y \in U) | f(x) - f(y) | < \epsilon$ .

P.S.$C(K, \mathbb{R}) $ is set of all continuous functions from $ K$ to $ \mathbb{R}$.

I'm not sure what technique should I use for solivng this. Any hint helps!

Answer 1

Assuming $K$ is Hausdorff and $d_\infty$ is the metric induced by the uniform norm $C(X,\Bbb R)$, the claim required to prove is a corollary of Arzelà–Ascoli theorem for compact Hausdorff spaces.