Kechris’ theorem

It’s not as wonderful as Marden’s, but this cute little result of Kechris can be really useful when you need to tweak a topology to get some extra functions to be continuous without disturbing the measurable structure.

Theorem. Let (XT) be a Polish space and Y a second-countable topological space. Let F be a countable collection of Borel-measurable functions fX → Y. Then there exists a topology T1 on X such that T1 ⊇ T, T1 is Polish, T and T1 generate the same Borel σ-algebra, and every f ∈ F is continuous with respect to T1.

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