Core-compact space

In general topology and related branches of mathematics, a '''core-compact''' topological space <math>X</math> is a topological space whose partially ordered set of open subsets is a continuous poset.<ref name="enofmath">{{cite encyclopedia|encyclopedia=Encyclopedia of mathematics|title=Core-compact space|url=https://encyclopediaofmath.org/wiki/Core-compact_space}}</ref> Equivalently, <math>X</math> is core-compact if it is exponentiable in the category Top of topological spaces.<ref name="enofmath"/><ref name="Gierz">{{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 }}</ref><ref name="nlabexp">{{nlab|id=exponential+law+for+spaces|title=Exponential law for spaces.}}</ref> This means that the functor <math> X\times - : \bf{Top} \to \bf{Top} </math> has a right adjoint. Equivalently, for each topological space <math> Y </math>, there exists a topology on the set of continuous functions <math> \mathcal{C}(X,Y) </math> such that function application <math>X \times \mathcal{C}(X, Y) \to Y</math> is continuous, and each continuous map <math> X\times Z \to Y</math> may be curried to a continuous map <math> Z \to \mathcal{C}(X,Y) </math>. Note that this is the Compact-open topology if (and only if)<ref name="MathOverflow"> {{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}}</ref> <math> X </math> 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 open neighborhood <math>U</math> of a point <math>x</math> contains an open neighborhood <math>V</math> of <math>x</math> that is way-below <math>U</math>; <math>V</math> is way-below (or relatively compact in) <math>U</math> if and only if every open cover containing <math>U</math> contains a finite subcover of <math>V</math>.<ref name="enofmath"/> As a result, every locally compact space is core-compact. For Hausdorff spaces (or more generally, sober spaces<ref name="Sotirov"> {{cite web|title=The compact-open topology: what is it really? |author=Vladimir Sotirov |url=https://wiki.math.wisc.edu/images/Compact-openTalk.pdf}}</ref>), 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.

== References ==  {{reflist}}

== Further reading == * {{cite web |url=https://math.stackexchange.com/q/1287458 |title=core-compact but not locally compact |date=June 20, 2016 |work=Stack Exchange }}

Category:Properties of topological spaces