Results 1 to 4 of 4

Thread: Quotient spce is Hausdorff

  1. #1
    Member
    Joined
    Feb 2009
    Posts
    103

    Quotient spce is Hausdorff

    Prove that a quotient topological space is Hausdorff if and only if any two distinct equivalence classes are contained in two disjoint open saturated sets.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    tah
    tah is offline
    Junior Member
    Joined
    Feb 2009
    Posts
    51
    Quote Originally Posted by Amanda1990 View Post
    Prove that a quotient topological space is Hausdorff if and only if any two distinct equivalence classes are contained in two disjoint open saturated sets.
    Let $\displaystyle \pi$ be the canonical projection, image by $\displaystyle \pi$ of an open saturated set of $\displaystyle X$ is open in the quotient topology. So the converse implication is obvious. For the direct implication, two distinct equivalence classes $\displaystyle [x_1],[x_2]$are contained in two disjoint open sets $\displaystyle O_1,O_2$, by definition $\displaystyle \pi^{-1}(O_i)$'s are open sets in $\displaystyle X$, which are indeed disjoint, saturated and $\displaystyle [x_i] \subseteq O_i $
    Last edited by tah; Feb 27th 2009 at 07:57 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    tah
    tah is offline
    Junior Member
    Joined
    Feb 2009
    Posts
    51
    Quote Originally Posted by tah View Post
    Let $\displaystyle \pi$ be the canonical projection, image by $\displaystyle \pi$ of an open saturated set of $\displaystyle X$ is open in the quotient topology. So the converse implication is obvious. For the direct implication, two distinct equivalence classes $\displaystyle [x_1],[x_2]$are contained in two disjoint open sets $\displaystyle O_1,O_2$, by definition $\displaystyle \pi^{-1}(O_i)$'s are open sets in $\displaystyle X$, which are indeed disjoint, saturated and $\displaystyle [x_i] \subseteq O_i $
    Sorry the last expression $\displaystyle [x_i] \subseteq O_i $ should be changed to $\displaystyle x_i \in \pi^{-1}(O_i) $.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Nov 2008
    Posts
    394
    This is a more detailed one that I've tried.

    Let X be a space and ~ an equivalence relation on X. Let X/~ denote the set of all equivalence classes $\displaystyle [x]=\{y \in X : x \sim y\} $ determined by ~. Let q be a quotient map of ~ such that $\displaystyle q:X \rightarrow X/\sim$. Now, the space $\displaystyle X/\sim$ induced by a quotient map q is a quotient space of X.

    Suppose a quotient topological space $\displaystyle X/\sim$ is Hausdorff.

    Let $\displaystyle [x], [y] $ be distinct points in $\displaystyle X/\sim$ and U, V be disjoint open sets in $\displaystyle X/\sim$ containing $\displaystyle [x]$ and $\displaystyle [y] $, respectively. By definition of a saturated set and Hausdorff property of $\displaystyle X/\sim$, $\displaystyle q^{-1}(U)$ and $\displaystyle q^{-1}(V)$ are disjoint open saturated sets containing $\displaystyle q^{-1}([x])$ and $\displaystyle q^{-1}([y])$ in X, respectively. The $\displaystyle q^{-1}([x])$ and $\displaystyle q^{-1}([y])$ are simply equivalence classes in X determined by ~.
    (If X/~ is Hausdorff, then X is Hausdorff. But the converse is not necessarily true).

    Conversely, suppose any two distinct equivalence classes $\displaystyle q^{-1}([x])$ and $\displaystyle q^{-1}([y])$ in X determined by ~ are contained in two disjoint open saturated sets V and W in X.

    Now, for any pair of distinct points $\displaystyle x \in q^{-1}([x])$ and $\displaystyle y \in q^{-1}([y])$, we have two distinct points q(x) and q(y) in $\displaystyle X/\sim$ contained in two distinct open sets q(V) and q(W), respectively (The q maps saturated open sets of X to open sets of X/~).
    Thus, $\displaystyle X/\sim$ is Hausdorff.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] A subspace of a Hausdorff space is Hausdorff
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: Sep 24th 2011, 01:40 PM
  2. Quotient map from Hausdorff to non-Hausdorff space
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Nov 28th 2009, 05:51 PM
  3. Which of these are Hausdorff and why?
    Posted in the Differential Geometry Forum
    Replies: 12
    Last Post: Nov 6th 2009, 03:57 AM
  4. Hausdorff
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Oct 23rd 2009, 04:39 AM
  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