I’m wondering if there is a version of Arzelà-Ascoli for continuous functions on not-necessarily compact metric/Hausdorff spaces $X$, i.e. a characterization of the compact subsets of $C_b(X)$ (under the sup norm) for $X$ not-necessarily compact, e.g. the titular $X=(0,1)$, using a neat set of conditions.
I assume there is not such a characterization, which brings me to perhaps the more important question: is there a philosophical reason or guiding intuition/heuristic that tells us that we “should not expect” such a theorem for non-compact spaces $X$?
Or dually is there a philosophical reason or guiding intuition/heuristic that tells us that we “should expect” such a theorem for compact spaces $X$? I know that continuous functions on compact sets enjoy a wide variety of nice properties (boundedness, attaining extrema, uniformity of continuity and convergence, etc.) but how do I know that these properties are “the best properties” and therefore indicative that continuous functions “really should belong on compact sets”/“live most naturally on compact sets”?
EDIT: there is a version of Arzelà-Ascoli for LCH spaces under compact subset convergence https://math.stackexchange.com/questions/20670/theorem-of-arzel%C3%A0-ascoli, but (1) that's not exactly what I am asking about and (2) compactness still permeates that entire result, i.e. it is clearly just a slight augmentation of the standard compact Arzelà-Ascoli. Some comments bring up inheriting an Arzelà-Ascoli theorem from a compactification, but that still leaves the question of why it is "more natural" to first do the compactness case and then try to extend from there.
(Cross-post of https://math.stackexchange.com/questions/4339552/arzel%c3%a0-ascoli-for-c-b0-1-or-more-generally-why-is-that-continuous-function).
EDIT: https://math.stackexchange.com/questions/35115/classifying-the-compact-subsets-of-lp tells us there is a nice characterization of compact subsets of $L^p(\mathbb R^n)$, and the Hanche-Olsen & Holden paper the answer links (https://arxiv.org/pdf/0906.4883.pdf) references an article of Phillips (Thm. 3.7 of https://www.ams.org/journals/tran/1940-048-03/S0002-9947-1940-0004094-3/) providing a characterization of compact subsets of any Banach space, including of course $C_b(X)$ for any topological space $X$. I am unsure as to the usefulness of Phillips' characterization.