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Math Help - Openness in quotient space

  1. #1
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    Openness in quotient space

    Let X be a topological space. I can prove that the set Homeo(X) of homeomorphisms f:X \rightarrow X becomes a group when endowed with the binary operation of composition.

    I can also show that if G is a subgroup of Homeo(X), then the relation "x ~ _{G} y iff there exists g \in G such that g(x) = y" is an equivalence relation.

    Now the question: Let G and ~ _{G} be as above, and let p: X \rightarrow X /~ _{G} be the quotient map. Prove that for every U open in X, P(U) is open in X / ~ _{G}.

    Let X = {R}^n \ {0}, n\geq 2, and let G be the subgroup of Homeo(X) composed of the maps g(x) = cx where c is a constant. Prove that x / ~ _{G} is the real projective space P {R}^{n-1}.
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  2. #2
    Senior Member
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    Nov 2008
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    Paris
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    Hi

    Let H be a group, and consider a group action of H on a topological space Y. Then the quotient map q:Y\rightarrow Y/H is open: indeed, since translations (given a h\in H, maps in Y^Y like y\mapsto h.y ) are homeomorphisms, given an open set U of  Y, q^{-1}(q(U))= \bigcup\limits_{h\in H} h.U is a union of open sets (the image of an open set under a homeomorphism is an open set) thus q^{-1}(q(U)) is open and by quotient topology definition, q(U) is open.

    Ok, that's a more general case, but it can give you an idea to solve your problem, the chosen group being particular.
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