Algebra Seminar talk
2026-10-30
Mohandes Grilj
The Open Mapping Theorem for Polish Groups
Abstract:
If we consider two topological spaces $X$ and $Y$ together with a continuous map $f : X \to Y$, a natural question arises: under what conditions does such a map preserve open sets?
This is rarely the case, and even for bijections it generally fails, as one learns in an introductory topology course.
Functional analysis provides an important positive result. If $X$ and $Y$ are Banach spaces and $f$ is a continuous, surjective, linear map, then $f$ does preserve open sets.
The Open Mapping Theorem for Polish groups is an analogue of this statement: if $X$ and $Y$ are Polish groups—that is, groups endowed with separable, completely metrizable topologies compatible with their group structures—and if $f$ is a continuous, surjective group homomorphism, then $f$ is an open map.