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Concept
Compactly generated space
Summary
In topology, a topological space is called a compactly generated space or k-space if its topology is determined by compact spaces in a manner made precise below. There is in fact no commonly agreed upon definition for such spaces, as different authors use variations of the definition that are not exactly equivalent to each other. Also some authors include some separation axiom (like Hausdorff space or weak Hausdorff space) in the definition of one or both terms, and others don't.
In the simplest definition, a compactly generated space is a space that is coherent with the family of its compact subspaces, meaning that for every set is open in if and only if is open in for every compact subspace Other definitions use a family of continuous maps from compact spaces to and declare to be compactly generated if its topology coincides with the final topology with respect to this family of maps. And other variations of the definition replace compact spaces with compact Hausdorff spaces.
Compactly generated spaces were developed to remedy some of the shortcomings of the . In particular, under some of the definitions, they form a while still containing the typical spaces of interest, which makes them convenient for use in algebraic topology.
Let be a topological space, where is the topology, that is, the collection of all open sets in
There are multiple (non-equivalent) definitions of compactly generated space or k-space in the literature. These definitions share a common structure, starting with a suitably specified family of continuous maps from some compact spaces to The various definitions differ in their choice of the family as detailed below.
The final topology on with respect to the family is called the k-ification of Since all the functions in were continuous into the k-ification of is finer than (or equal to) the original topology . The open sets in the k-ification are called the in they are the sets such that is open in for every in Similarly, the in are the closed sets in its k-ification, with a corresponding characterization.
Official source
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