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Most topological spaces commonly studied in mathematics are compactly generated. For instance, every Locally Compact (and Compact ) space is compactly generated, as is every First-countable Space . Additionally, every CW Complex is compactly generated. One of the primary motivations for studying compactly generated spaces comes from Category Theory . The Category Of Topological Spaces , Top, is defective in the sense that it fails to be a Cartesian Closed Category . There have been various attempts to remedy this situation, one of which is to restrict oneself to the Full Subcategory of '''compactly generated Hausdorff spaces''', i.e. compactly generated spaces which are also Hausdorff . This category is, in fact, cartesian closed. A definition of the exponential object is given below. The category of compactly generated Hausdorff spaces is general enough to include all Metric Space s, Topological Manifold s, and all CW Complex es. PROPERTIES Given any topological space ''X'' we can define a (possibly) Finer Topology on ''X'' which is compactly generated. Let {''K''α} denote the family of compact subsets of ''X''. We define the new topology on ''X'' by declaring a subset ''A'' to be closed Iff ''A'' ∩ ''K''α is closed in ''K''α for each α. Denote this new space by ''X''c. One can show that the compact subsets of ''X''c and ''X'' coincide and the induced topologies are the same. It follows that ''X''c is compactly generated. If ''X'' was compactly generated to start with then ''X''c = ''X'' otherwise the topology on ''X''c is strictly finer than ''X'' (i.e. there are more open sets). This construction is Functorial . The functor from Top to '''CGHaus''' which takes ''X'' to ''X''c is Left Adjoint to the Forgetful Functor '''CGHaus''' → Top. The it is continuous when restricted to each compact subset ''K'' ⊆ ''X''. If ''X'' and ''Y'' are two compactly generated spaces the Product ''X'' × ''Y'' may not be compactly generated (it will be if at least one of the factors is locally compact). Therefore when working in categories of compactly generated spaces it is necessary to define the product as (''X'' × ''Y'')c. The Exponential Object in the CGHaus is given by (''Y''''X'')c where ''Y''''X'' is the space of Continuous Map s from ''X'' to ''Y'' with the Compact-open Topology . SEE ALSO |
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