Intersections of compactly many open sets are open

Martín Hötzel Escardó

Published 2020-01-16Version 1

By definition, the intersection of finitely many open sets of any topological space is open. Nachbin observed that, more generally, the intersection of compactly many open sets is open. Moreover, Nachbin applied this to obtain elegant proofs of various facts concerning compact sets in topology and elsewhere. A simple calculation shows that Nachbin's observation amounts to the well known fact that if a space $X$ is compact, then the projection map $Z \times X \to Z$ is closed for every space $Z$. It is also well known that the converse holds: if a space $X$ has the property that the projection $Z \times X \to Z$ is closed for every space $Z$, then $X$ is compact. We reformulate this as a converse of Nachbin's observation, and apply this to obtain further elegant proofs of (old and new) theorems concerning compact sets. We also provide a new proof of (a reformulation of) the fact that a space $X$ is compact if and only if the projection map $Z \times X \to Z$ is closed for every space $Z$. This is generalized in various ways, to obtain new results about spaces of continuous functions, proper maps, relative compactness, and compactly generated spaces. In particular, we give an intrinsic description of the binary product in the category of compactly generated spaces in terms of the Scott topology of the lattice of open sets.