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 s ...

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 ...

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 invariants that classify topological spaces up to homeomorphism, though usually most classify ...

as a substitute for the category of topological spaces, 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 subsp ...

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

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 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 id ...

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