Core-compact space

In general topology and related branches of mathematics, a core-compact topological space $X$ is a topological space whose partially ordered set of open subsets is a continuous poset.{{cite encyclopedia|encyclopedia=Encyclopedia of mathematics|title=Core-compact space|url=https://encyclopediaofmath.org/wiki/Core-compact_space}} Equivalently, $X$ is core-compact if it is exponentiable in the category Top of topological spaces.{{cite book|author-link6=Dana Scott|date=2003|doi=10.1017/CBO9780511542725|first1=Gerhard|first2=Karl|first3=Klaus|first4=Jimmie|first5=Michael|first6=Dana S.|isbn=978-0-521-80338-0|language=en|last1=Gierz|last2=Hofmann|last3=Keimel|last4=Lawson|last5=Mislove|last6=Scott|location=Cambridge|mr=1975381|publisher=Cambridge University Press|series=Encyclopedia of Mathematics and Its Applications|title=Continuous lattices and domains|volume=93|zbl=1088.06001|s2cid=118338851 }}{{nlab|id=exponential+law+for+spaces|title=Exponential law for spaces.}} This means that the functor

$X\backslash times\; -\; :\; \backslash bf\{Top\}\; \backslash to\; \backslash bf\{Top\}$

has a right adjoint. Equivalently, for each topological space $Y$, there exists a topology on the set of continuous functions

$\backslash mathcal\{C\}(X,Y)$ such that function application

$X\; \backslash times\; \backslash mathcal\{C\}(X,\; Y)\; \backslash to\; Y$ is continuous, and each continuous map

$X\backslash times\; Z\; \backslash to\; Y$ may be curried to a continuous map

$Z\; \backslash to\; \backslash mathcal\{C\}(X,Y)$.

Note that this is the Compact-open topology if (and only if)

{{cite web|title

=Exponential law w.r.t. compact-open topology

|author=Tim Campion

|url=https://mathoverflow.net/questions/307493/exponential-law-w-r-t-compact-open-topology}}

$X$ is locally compact. (In this article locally compact means that every point has a neighborhood base of compact neighborhoods; this is definition (3) in the linked article.)

Another equivalent concrete definition is that every neighborhood $U$ of a point $x$ contains a neighborhood $V$ of $x$ whose closure in $U$ is compact. As a result, every locally compact space is core-compact. For Hausdorff spaces (or more generally, sober spaces

{{cite web|title

=The compact-open topology: what is it really?

|author=Vladimir Sotirov

|url=https://wiki.math.wisc.edu/images/Compact-openTalk.pdf}}), core-compact space is equivalent to locally compact. In this sense the definition is a slight weakening of the definition of a locally compact space in the non-Hausdorff case.