Recall that a space is locally compact if each point has a neighborhood basis consisting of compact neighborhoods.
Somewhat less known but slightly more general is the class of spaces called “core-compact spaces”:
A topological space is core-compact if, for every open neighborhood of a point , there is a smaller open neighborhood of , such that every open covering of admits a finite subcover of . We write (pronounced “ is well below ”) to indicate that this covering condition holds.
Core-compactness is equivalent to the statement that every open is the union of open sets such that .
The goal of this article is to give a succinct proof that core-compact spaces are exponentiable in , so that in particular locally compact spaces (with no further regularity conditions) are exponentiable in . This proof is extracted from a nice article by Escardó and Heckmann, who show moreover that exponentiable spaces in are precisely the core-compact spaces (which is a matter we won’t go into here).
For given topological spaces and , we let be the set of continuous functions . The theory developed by Escardó and Heckmann shows that there is a “best” candidate for an exponential topology on , which they call the Isbell topology but which is also known as the “natural topology” or the “topology of continuous convergence”. Whatever it is called, it is uniquely characterized by the following property:
In the case where is the Sierpinski space (where is open but is not), is in natural bijection with the set of open sets of , i.e., the topology of . Under this isomorphism the natural topology on corresponds to a topology also known as the Scott topology on :
A subset is called Scott-open if
it is upward-closed: if are open sets of and , then ;
for every there exists such that .
Then for arbitrary spaces , the natural topology on may be explicitly described as the topology generated by sets of the form
where ranges over open sets of and over Scott-open subsets of .
We let denote the space whose underlying set is and whose topology is the natural topology thus described. The theorem we prove may be stated as follows:
If is core-compact and is any space, then the evaluation map is continuous, and realizes (via the Yoneda lemma) a natural isomorphism . Thus, satisfies the universal property of an exponential in .
As we said, the proof we give is a very whittled-down rendition extracted from the much more thorough account given by Escardó and Heckmann. It is based on two lemmas:
For any spaces , and (without assuming is locally compact), if is continuous, then its transpose is continuous.
For any space , the evaluation map is continuous if is core-compact.
It follows that if is continuous, then so is the inverse transpose
As the operations and are mutually inverse, this establishes the natural isomorphism of the theorem.
First we establish Lemma 1.
Suppose is continuous, and let be a typical generating element of the topology of . We are to show that
is open in . Fix an element ; then
Consider the collection of all pairs where is open in and is an open neighborhood of such that . The collection of such covers . Since and is Scott-open, there are finitely many such , say , that cover some , and we have corresponding open neighborhoods of such that for .
We claim the neighborhood of the point is included in . Indeed, suppose ; we must show . But since is Scott-open, it is upward closed, and so it suffices to verify that
But if , then for some and of course , so and the claim is proven.
Now we prove Lemma 2.
Let ; let be an open neighborhood of . Then is an open neighborhood of , and since is core-compact, there is an open neighborhood of such that . Put
Then it is enough to show
is Scott-open (and hence is open in );
;
.
The only tricky part is 1. For 2. it is obvious that since , i.e., . For 3., if and , then , whence and then , as required.
For 1., it is obvious that if and , then , so is upward-closed. Now we need to check that for all in there exists in such that . By core-compactness,
The covering of by open sets such that is a collection that is closed under finite unions, and so since , we have that for some such . If , then , so indeed there is such that . This completes the proof that is Scott-open and the proof of the lemma.
It is not hard to deduce that if is core-compact, then the collections of the form
form a base of the Scott topology on . In that case, the exponential = natural topology on may be described as generated by sets of the form
where range over open sets of , respectively. This bears a closer resemblance to the classical compact-open topology than our first abstract description of the natural topology on in terms of the Scott topology on .