Arzelà-Ascoli for $C_b(0,1)$? Or more generally, why is that continuous functions "live most naturally" on compact spaces? 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.
 A: A possible proof of the Arzelà-Ascoli theorem for $C(X)$ with compact $X$ is to show that equicontinuity of a subset $\mathcal F\subseteq C(X)$ implies that the topology of pointwise convergence coincides on $\mathcal F$ with the topology of the sup-norm. That $\mathcal F$ is relatively compact for the pointwise topology follows from Tychonov's theorem. For a general topological space and the space $C_b(X)$ this proof gives an abstract characterization:
A subset $\mathcal F$ of $C_b(X)$ is compact for the sup-norm $\|\cdot\|$ if and only if it is closed and bounded and, for all $f\in\mathcal F$ and $\varepsilon>0$, there are $\delta>0$ and a finite subset $E\subseteq X$ such that, for every $g\in \mathcal F$ with $\max\{|f(x)-g(x)|:x\in E\}<\delta$ we have $\|f-g\|<\varepsilon$.
It depends very much on the concrete situation whether this is useful. As an example, the general result implies that a closed and bounded subset $\mathcal F\subseteq C_b(X)$ for a precompact metric space $X$ is compact if it is equi-uniformly continuous, i.e., for every $\varepsilon>0$ there is $\delta>0$ such that, for all $f\in\mathcal F$ and $x,y\in X$ with $d(x,y)<\delta$, one has $|f(x)-f(y)|<\varepsilon$.
Indeed, given $f$ and $\varepsilon$ and the corresponding $\delta$ there is a finite set $E\subseteq X$ such that every $x\in X$ is $\delta$-close to some $x'\in E$. If then $\max\{|f(y)-g(y)|:y\in E\}<\varepsilon$ one gets, for every $x\in X$, that $|f(x)-g(x)|\le |f(x)-f(x')|+|f(x')-g(x')|+|g(x')-g(x)|< 3\varepsilon$. This gives $\|f-g\|\le 3\varepsilon$ where, of course, the factor $3$ is inessential.
I know that this "application" can also be reduced to the classical Arzelà-Ascoli theorem by extending all uniformly continuous functions to the completion of $X$ which is compact.
A: The Azrela Ascoli theorem is true on any space $C^0(Y,Z)$ with $Z$ metric and $Y$ an arbitrary topological space, as soon as $C^0(Y,Z)$ is endowed with the compact open topology. See e.g. Dugundgi, Topology, chap. XII Theorem 6.4.
