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The density topology is not generated. (English) Zbl 0732.26004

A topological space (X,t) is generated if for each topology \(t'\) such that the set of continuous functions \(f: (X,t')\to (X,t')\) contains the set of all continuous functions \(f: (X,t)\to (X,t)\) we have also \(t'\supset t\). Using the characterization of generated spaces given by J. C. Warndorf [Fundam. Math. 66, 25-43 (1969; Zbl 0189.232)] the authors have proved that the density topology is not generated.

MSC:

26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
26A30 Singular functions, Cantor functions, functions with other special properties
26A21 Classification of real functions; Baire classification of sets and functions

Citations:

Zbl 0189.232
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