Concept

Compactly generated space

Summary
In topology, a topological space X 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 A \subseteq X, A is open in X if and only if A \cap K is open in K for every compact subspace K \subseteq X. Other definitions use a family of continuous maps from compact spaces to X and declare X to be compactly generated if its topolog
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