English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

By definition, the intersection of finitely many open sets of any topologicalspace is open. Nachbin observed that, more generally, the intersection ofcompactly many open sets is open. Moreover, Nachbin applied this to obtainelegant proofs of various facts concerning compact sets in topology andelsewhere. A simple calculation shows that Nachbin's observation amounts to thewell 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 theconverse 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 thisas a converse of Nachbin's observation, and apply this to obtain furtherelegant proofs of (old and new) theorems concerning compact sets. We alsoprovide a new proof of (a reformulation of) the fact that a space $X$ iscompact if and only if the projection map $Z \times X \to Z$ is closed forevery space $Z$. This is generalized in various ways, to obtain new resultsabout spaces of continuous functions, proper maps, relative compactness, andcompactly generated spaces. In particular, we give an intrinsic description ofthe binary product in the category of compactly generated spaces in terms ofthe Scott topology of the lattice of open sets.

Intersections of compactly many open sets are open