Using the topology of uniform convergence for functions over non-compact spaces Let $(X, d)$ be a (complete) metric space, and $C(X)$ be the space of continuous maps over $X$. If $X$ is compact, one often uses the topology of uniform convergence when analyzing $C(X)$. If $X$ is non-compact, there are a bunch of other topologies that one can attach to $C(X)$: the compact-open topology, the strong topology, ...
My question is the following: Why don't we just use the topology of uniform convergence on $C(X)$ regardless of the compactness of $X$? Generally, the first answer I get to this question is that the metric $d(f, g) = \sup_{x \in X} d(f(x), g(x))$ becomes unbounded when $X$ is not compact. Why is that a problem? We can always consider it an extended metric, and all the theory carries over without any problems. Even if infinite-valued metrics turn out to break the theory somehow, we can always use $d'(f, g) \equiv d(f, g)(1 + d(f, g))^{-1}$.
Therefore, I am suspecting that the topology of uniform convergence has some other disadvantage(s) when treating functions over non-compact spaces. What are these disadvantages? What concern(s) made mathematicians invent the other topologies? Why is this topology not used commonly?
 A: A few things that come to mind:


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*The topology of uniform convergence is too fine on e.g. $C(X)$ where $X$ is an open subset of some $\mathbb{R}^n$. In many situations, you don't have uniform convergence, but locally uniform convergence, and the locally uniform convergence is sufficient for many theorems (the limit of a locally uniformly convergent sequence of continuous functions is continuous; interchange of limit and integral for integrals over compact subsets [path or surface integrals]), and when it isn't, globally uniform convergence is often also not sufficient (integrals over sets of infinite measure).

*With the topology of uniform convergence, $C(X)$ is not a topological vector space if there are unbounded continuous functions on $X$ (scalar multiplication is not continuous), nor is it a topological ring then.
The topology of uniform convergence is a natural topology for the space $C_b(X)$ of bounded continuous functions, however. Note that when $X$ is compact, we have $C(X) = C_b(X)$.
Still, for some purposes, the topology of uniform convergence can be an/the adequate topology on $C(X)$ to consider for non-compact $X$, but there are purposes where it is inadequate, and hence other topologies have to be considered.
