Results 1 to 2 of 2

Thread: Openness in quotient space

  1. #1
    Newbie
    Joined
    Feb 2009
    Posts
    11

    Openness in quotient space

    Let X be a topological space. I can prove that the set Homeo(X) of homeomorphisms $\displaystyle f:X \rightarrow X$ 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 ~$\displaystyle _{G}$ y iff there exists $\displaystyle g \in G$ such that $\displaystyle g(x) = y$" is an equivalence relation.

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

    Let $\displaystyle X = {R}^n \ {0}, n\geq 2$, and let G be the subgroup of Homeo(X) composed of the maps $\displaystyle g(x) = cx$ where c is a constant. Prove that $\displaystyle x /$ ~$\displaystyle _{G}$ is the real projective space $\displaystyle P {R}^{n-1}$.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    From
    Paris
    Posts
    354
    Hi

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

Similar Math Help Forum Discussions

  1. Quotient Space
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: Oct 6th 2011, 05:44 PM
  2. Quotient space
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Sep 28th 2011, 01:20 PM
  3. quotient space
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Apr 25th 2010, 07:51 PM
  4. Topology Quotient Space
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: May 17th 2008, 01:42 PM
  5. Non-Hausdorff Quotient Space
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Oct 12th 2006, 04:30 PM

Search Tags


/mathhelpforum @mathhelpforum