Automatic continuity of measurable homomorphisms on Cech-complete topological groups

Taras Banakh

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arxiv 2206.02481 v3 pith:XHN5L35H submitted 2022-06-06 math.GN math.GR

Automatic continuity of measurable homomorphisms on Cech-complete topological groups

Taras Banakh This is my paper

classification math.GN math.GR

keywords continuitygroupshomomorphismsmeasurableonlytopologicalcech-completecompact

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We prove that a homomorphism $h:X\to Y$ from a (locally compact) Cech-complete topological group $X$ to a topological group $Y$ is continuous if and only if $h$ is Borel-measurable if and only if $h$ is universally measurable (if and only if $h$ is Haar-measurable). This answers a problem of Kuznetsova and extends a result of Kleppner on the continuity of Haar-measurable homomorphisms between locally compact groups and a result of Rosendal on the continuity of universally measurable homomorphisms between Polish groups.

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Polish topologies on endomorphism monoids of linear orders
math.GR 2026-05 unverdicted novelty 6.0

Authors introduce property XX for automatic continuity and prove that End^∞(N,≤) has a unique Polish semigroup topology while End(N,≤), End(Z,≤) have infinitely many and End(N,<) has continuum many.