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Math Help - separation axioms

  1. #1
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    separation axioms

    prove that :
    Every  \displaystyle \tau \scriptstyle3 space is \displaystyle \tau \scriptstyle 2 \frac{1}{2}
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  2. #2
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    A good way to start: write out the definitions of T_3 and T_{2 1/2}
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  3. #3
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    Quote Originally Posted by flower3 View Post
    prove that :
    Every  \displaystyle \tau \scriptstyle3 space is \displaystyle \tau \scriptstyle 2 \frac{1}{2}
    Definition. A space X is a T_{2\frac{1}{2}}-space provided that for each pair x,y of distinct points of X, there exists open sets U and V with disjoint closures such that x \in U and y \in V .

    Lemma 1. X is regular Hausdorff ( T_3) if and only if given a point x of X and a neighborhoood U of x, there is a neighborhood W of x such that \bar{W} \subset U.

    Assume X is regular Hausdorff ( T_3). Since X is also Hausdorff, for each pair x,y of distinct points of X, there exists disjoint open sets U and V containing x and y, respectively. By hypothesis, it follows that there is a neighborhood W of x such that \bar{W} \subset U and a neighborhood Y of y such that \bar{Y} \subset V by lemma 1. Since U and V are disjoint, we see that \bar{W} and \bar{Y} are disjoint. Now, for each pair x,y of distinct points of X, there exists open sets W and Y with disjoint closures such that x \in W and y \in Y .Thus, X is T_{2\frac{1}{2}}-space.

    The converse ("Every T_{2\frac{1}{2}}-space is T_3") is not necessarily true. The example where a space X is T_{2\frac{1}{2}}-space but not T_3 can be found in the K-topology on \mathbb{Re}, where a closed set K=\{1/n | n \in \mathbb{Z}^+\} (Note that K is closed in the K-topology on \mathbb{Re} ) and {0} cannot be separated by disjoint open sets.
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