In mathematics, the category of

compactly generated In mathematics, compactly generated can refer to: *Compactly generated group, a topological group which is algebraically generated by one of its compact subsets *Compactly generated space In topology, a compactly generated space is a topological sp ...

weak Hausdorff space In mathematics, a weak Hausdorff space or weakly Hausdorff space is a topological space where the image of every continuous map from a compact Hausdorff space into the space is closed. In particular, every Hausdorff space is weak Hausdorff. As a ...

s CGWH is one of typically used categories in

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

as a substitute for the

category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again contin ...

, as the latter lacks some of the pleasant properties one would desire. There is also such a category for based spaces, defined by requiring maps to preserve the base points. The articles

compactly generated space In topology, a compactly generated space is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space ''X'' is compactly generated if it satisfies the following condition: :A subspa ...

and

define the respective topological properties. For the historical motivation behind these conditions on spaces, see Compactly generated space#Motivation. This article focuses on the properties of the category.

Properties

CGWH has the following properties: *It is

complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...

and cocomplete. *The forgetful functor to the sets preserves small limits. *It contains all the locally compact Hausdorff spaces and all the

CW complexes A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...

. *The

internal Hom In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and ...

exists for any pairs of spaces ''X'', ''Y''; it is denoted by

\operatorname(X, Y)

Y^X

and is called the (free) mapping space from ''X'' to ''Y''. Moreover, there is a homeomorphism *:

\operatorname(X \times Y, Z) \simeq \operatorname(X, \operatorname(Y, Z))

:that is natural in ''X'', ''Y'', ''Z''. In short, the category is

Cartesian closed In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in math ...

in an enriched sense. *A finite product of CW complexes is a CW complex. *If ''X'', ''Y'' are based spaces, then the

smash product In topology, a branch of mathematics, the smash product of two pointed spaces (i.e. topological spaces with distinguished basepoints) (''X,'' ''x''0) and (''Y'', ''y''0) is the quotient of the product space ''X'' × ''Y'' under the ide ...

of them exists. The (based) mapping space

\operatorname(X, Y)

from ''X'' to ''Y'' consists of all base-point-preserving maps from ''X'' to ''Y'' and is a closed subspace of the mapping space between the underlying unbased spaces. It is a based space with the base point the unique constant map. For based spaces ''X'', ''Y'', ''Z'', there is a homeomorphism *:

\operatorname(X \wedge Y, Z) \simeq \operatorname(X, \operatorname(Y, Z))

:that is natural in ''X'', ''Y'', ''Z''.

Notes

References

* * *

Properties

Notes

References

Further reading