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canonical injection canonical quotient category of T2 closed embedding closed in FX closed subset cocomplete coequalizers colimits compact subsets completely regular T2 continuous functions continuous injection coproduct Corollary countable define denote dense range determine the topology diagonally separable DISSERTATIONES MATHEMATICAE example exists fc-embedding fc-groups are closed fc-product fc-refinement fc-topology free group free product functionally T2 FX is functionally FX is T2 G in KG G is t2 grounding functor H is closed Hausdorff separation hence identity neighborhood inversion k-embedding k-group K-left k-space left adjoint Lemma locally compact spaces morphism multiplication on G non-Hausdorff normal fc-space normal subgroup nr(A open map positive integer Proof Proposition quasi-compact space quotient in KG quotient topology induced subspace topology surjection T-+X T2 fc-spaces t2 quotient t2 separation t2 spaces test maps TOP(X topological embedding topological group topological isomorphism topological product topological space trivial weakly Hausdorff X-+Y