I quote page 285 of Munkres:

The definitions of the uniform topology and the compact convergence topology made specific use of the metric $d$ for the space $Y$. But the topology of pointwise convergence did not; in fact, it is defined for any space $Y$. It is natural to ask whether either of these other topologies can be extended to the case where $Y$ is an arbitrary topological space. There is no satisfactory answer to this question for the space $Y^X$ of all functions mapping $X$ to $Y$. But for the subspace of continuous functions one can prove something. It turns out that there is in general a topology on $\mathcal{C}(X,Y)$, called the compact open topology, that coincides with the compact convergence topology when $Y$ is a metric space.

I would like to have some idea of why such an attempt on non-continuous functions might fail. Take the compact open topology for instance, which defines open sets to be the set of functions which sends compact subspaces into open sets in the domain. When $Y$ is a metric space, it is easy to describe basis of open sets - they are just $\epsilon$-balls. One would then find it natural to replace the $\epsilon$- ball with an arbitrary open set for a general topological space, and this intuitively allows us to capture 'enough' continuous functions as small pieces of the continuous functions will not lie too far away from this open ball.

So why can't we make this approach work for discontinuous functions as well, say, by defining a topology which generalizes the uniform topology to one in which an open set is defined to be the set of functions which map into an open set of the Box topology on $Y$ (or if that is too badly behaved, why not those functions which map into $U^X$, where $U$ is an open set in $Y$.

What I'm saying is not very precise, hopefully someone could make the picture clearer for me. Thanks!

  • $\begingroup$ I think the question should be not whether you get a topology, rather whether you can do anything with the topology you get. For instance, in some sense the "natural" topology on a space $X$, at least from a categorical point of view, is the one that is compactly generated. $\endgroup$ – Matematleta Jul 8 '17 at 19:15
    I have some suggestions and quaries as well. Let X and Y be arbitrary topological spaces and F(X,Y) the set of all functions from X to Y. Let P(X) denote the power set of X. For any subcollection A(X) of P(X), let N(A,V)={f∈F(X,Y):f(A)⊆V}, where A∈A(X) and V an open set in Y. Then the collection {N(A,V):A∈A(X), open V⊆Y} forms a subbase for a "set-open" topology on F(X,Y), called the A(X)-open topology (following Arens-Dugundji (PJM, 1951, p. 13; see also the references, e.g., Fox(1945), Arens (1946), Arens-Dugundji (1951), Kelley (1955), Hu (1964), McCoy-Ntantu (1988)).
Let F(X), K(X), σK(X), σo(X) denote, respectivey, the collection of all finite, compact, σ-compact, countable subsets of X. Clearly, F(X)⊆ K(X)⊆σK(X)⊆P(X) and F(X)⊆σo(X)⊆σK(X), but in general σo(X) and K(X) are different.
Using the above terminology, we can consider the following "set-open" topologies on F(X,Y): 
(1) F(X)-open topology, the usual point-open topology, written briefly as Sp; (2) K(X)-open topology, the usual compact-open topology, written briefly as Sk; (3) σK(X)-open topology, the σ-compact-open topology, written briefly as Sσ; (4) σo(X)-open topology, the countable-open topology, written briefly as Sσo; (5) P(X)-open topology, written briefly as S_{P(X)}; (6) {X}-open topology, written briefly as S_{X}. It seems clear that these topologies are well-defined and that (a) Sp≤ Sk ≤ Sσ ≤ S_{P(X)} and Sp ≤ Sσo ≤ Sσ on F(X,Y). 
If we confine ourselves only to "set-open" topologies (not necessarily "uniform" type topologies), these topologies seem to be well-defined. But (except the point-open and compact-open topologies) I wonder why these types of topologies on C(X,Y) and F(X,Y) have not been explored much. However, it has been mentioned by Professor R. A. McCoy that the S_{X}-topology is in general (i) not even a Hausdorff topology, (ii) also not finer than or equal the compact-open topology Sk, so it would not be of much use in most settings. Perhaps, someone among the readers can shed more light on it. 
  • $\begingroup$ I appreciate and agree with your your point of view, Chilango. $\endgroup$ – Liaqat Ali Khan Jul 8 '17 at 20:35

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