A topological space X is compactly generated (a k-space) if its topology is determined by compact subspaces: a set A ⊆ X is closed (open) in X iff A ∩ K is closed (open) in K for every compact K ⊆ X, equivalently X carries the final topology with respect to inclusions of its compact subspaces.

Principle

Reduce global topological verification to checks on compact subsets so that constructions (notably function spaces with the compact-open topology) behave well categorically; compact subsets generate the topology in the sense of the final/quotient topology from them.

Demonstration

CW complexes and locally compact Hausdorff spaces are compactly generated; the category of compactly generated weak Hausdorff spaces is a standard convenient category in algebraic topology where products, mapping spaces and colimits have expected properties.

Misapplication

Confusing compactly generated with locally compact or sequential: a space can be compactly generated without being locally compact, and failing to check the compact-generation hypothesis can make exponential laws for function spaces fail.

Consequence

Using compactly generated spaces ensures that formation of mapping spaces and many categorical constructions (products, colimits) commute with expected operations, resolving pathologies of the naive category of all topological spaces and yielding a better-behaved homotopy theory.

Reversal

In the opposite situation, topologies not determined by compacts may force maps to be continuous on each compact subset but not continuous globally; such spaces break convenient categorical properties and complicate function-space constructions.

Boundary

Compact generation is a condition about how the topology is generated and does not imply Hausdorff, local compactness or metrizability by itself; variants (weak Hausdorff, k-spaces, k-compactness) specify additional separation or compactness constraints.

Semantic Tension

Semantic tension appears between compactly generated spaces and other 'convenient' categories (e.g., sequential spaces, locally compact spaces): each choice trades off different closure properties and technical conveniences for homotopy-theoretic constructions.

Synthesis

A compactly generated space is one whose topology can be tested on compact subsets: by requiring continuity and closure to be checked on compacts, the class of such spaces forms a convenient category in which mapping spaces and homotopical constructions behave coherently.