Theorem of Arzelà-Ascoli

I think this question has been already been answered through the helpful comments. So thanks to Henno Brandsma and t.b.! This is just to finally tick it off.

My conclusion: It seems that $X$ being Hausdorff is rather a matter of convenience (maybe to avoid issues with the definition of local compactness for non-Hausdorff spaces, as pointed out in the comments), than a necessary condition.

Also this version of the theorem seems quite general enough for most uses.

Why is Euler's Totient function always even?

Prove that the product of four consecutive positive integers plus one is a perfect square

how to show convergence in probability imply convergence a.s. in this case?

How would one go about proving that the rationals are not the countable intersection of open sets?

The ring $\Bbb Z\left [\frac{-1+\sqrt{-19}}{2}\right ]$ is not a Euclidean domain

Derived subgroup where not every element is a commutator

Group of positive rationals under multiplication not isomorphic to group of rationals

Geometric meaning of the determinant of a matrix

There exists only two groups of order $p^2$ up to isomorphism.

Proof of the independence of the sample mean and sample variance

Prove that the inverse image of an open set is open