general topology In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differ ...

and related branches of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a core-compact

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

X

is a

whose

partially ordered set In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...

open subsets In mathematics, an open set is a generalization of an open interval in the real line. In a metric space (a set with a distance defined between every two points), an open set is a set that, with every point in it, contains all points of the metr ...

is a

continuous poset In order theory, a continuous poset is a partially ordered set in which every element is the directed set, directed supremum of elements approximating it. Definitions Let a,b\in P be two elements of a preordered set (P,\lesssim). Then we say tha ...

. Equivalently,

X

is core-compact if it is exponentiable in the category Top of topological spaces. This means that the functor

X\times - : \bf \to \bf

has a right adjoint. Equivalently, for each topological space

Y

, there exists a topology on the set of continuous functions

\mathcal(X,Y)

such that

function application In mathematics, function application is the act of applying a function to an argument from its domain so as to obtain the corresponding value from its range. In this sense, function application can be thought of as the opposite of function abs ...

X \times \mathcal(X, Y) \to Y

is continuous, and each continuous map

X\times Z \to Y

may be curried to a continuous map

Z \to \mathcal(X,Y)

. Note that this is the

Compact-open topology In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory ...

if (and only if)

X

locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...

. (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

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 ), 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.

See also

References

Further reading