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Math Help - Quotient spce is Hausdorff

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    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.
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  2. #2
    tah
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    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 \pi be the canonical projection, image by \pi of an open saturated set of X is open in the quotient topology. So the converse implication is obvious. For the direct implication, two distinct equivalence classes [x_1],[x_2]are contained in two disjoint open sets O_1,O_2, by definition \pi^{-1}(O_i)'s are open sets in X, which are indeed disjoint, saturated and [x_i] \subseteq O_i
    Last edited by tah; February 27th 2009 at 08:57 AM.
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  3. #3
    tah
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    Quote Originally Posted by tah View Post
    Let \pi be the canonical projection, image by \pi of an open saturated set of X is open in the quotient topology. So the converse implication is obvious. For the direct implication, two distinct equivalence classes [x_1],[x_2]are contained in two disjoint open sets O_1,O_2, by definition \pi^{-1}(O_i)'s are open sets in X, which are indeed disjoint, saturated and [x_i] \subseteq O_i
    Sorry the last expression [x_i] \subseteq O_i should be changed to x_i \in \pi^{-1}(O_i) .
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    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 [x]=\{y \in X : x \sim y\} determined by ~. Let q be a quotient map of ~ such that q:X \rightarrow X/\sim. Now, the space X/\sim induced by a quotient map q is a quotient space of X.

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

    Let [x], [y] be distinct points in X/\sim and U, V be disjoint open sets in X/\sim containing [x] and  [y] , respectively. By definition of a saturated set and Hausdorff property of X/\sim, q^{-1}(U) and q^{-1}(V) are disjoint open saturated sets containing q^{-1}([x]) and q^{-1}([y]) in X, respectively. The q^{-1}([x]) and 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 q^{-1}([x]) and 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 x \in q^{-1}([x]) and y \in q^{-1}([y]), we have two distinct points q(x) and q(y) in 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, X/\sim is Hausdorff.
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