If a group $G$ acts on a topological space $X$ by homeomorphisms, why is the quotient map $X \to X/G$ necessarily an open map?
Quotient by Homeomorphisms Produces an Open Map
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general-topology
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0@HenningMakholm, the definition of the quotient topology I use is the second one provided on wikipedia [http://en.wikipedia.org/wiki/Quotient_space#Definition ] (begins with "Equivalently, we can define them..."). I see that with the first definition there, the openness of the quotient map is, indeed, immediate. Thank you! – 2012-02-04
1 Answers
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Let $q$ be the quotient map, and let $U$ be a non-empty open set in $X$; $q[U]$ is open in $X/G$ iff $q^{-1}[q[U]]$ is open in $X$. But $q^{-1}[q[U]]=\bigcup_{g\in G}(g\cdot U)$ is a union of open sets, since each map $x\mapsto g\cdot x$ is a homeomorphism, and therefore open itself, as desired.