  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.
