Let $X$ be a uniform topological space and $C([0,1],X)$ the space of continuous functions from [0,1] to $X$. Assume that for subsets of $X$ sequential compactness and compactness are equivalent. Let $(f_n)_{n\in \mathbb{N}} \subseteq C([0,1],X)$ satisfy the conditions of a suitably general version of the Arzela-Ascoli theorem (say, the sequence is uniformly equi-continuous, and for each $t$ the sequence $(f_n(t))_{n\in \mathbb{N}}$ is precompact in $X$). It follows that the closure of $(f_n)_{n\in \mathbb{N}}$ is compact in $C([0,1],X)$, with the topology of uniform convergence.
Question: Is there a converging subsequence w.r.t. the topology of uniform convergence?
Of course, this is true if $X$ a metric space. In the present situation, this question might be equivalent to the question whether the equivalence of compactness and sequential compactness is inherited from $X$ to $C([0,1],X)$ (more precisely, whether the implication compactness $\implies$ seq. compactness is inherited).
Alternatively, I would be interested whether the Arzela-Ascoli theorem, in the general situation posed above, holds if both the assumption and the assertion are "seq. (pre-)compactness" instead of "(pre-)compactness?
Thanks!
