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Math Help - Hausdorff and Compact Space

  1. #1
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    Hausdorff and Compact Space

    Let X be an infinite set and p a point in X, chosen once and for all. Let T be the collection of subsets V of X, for which either p is not a member of V, or p is a member of V and its complement ~V is finite.

    I have proved that T is a topology on X, but I don't seem to be able to prove that;

    a) (X,T) is a Hausdorff space.
    b) (X,T) is a compact space.
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    Quote Originally Posted by Cairo View Post
    Let X be an infinite set and p a point in X, chosen once and for all. Let T be the collection of subsets V of X, for which either p is not a member of V, or p is a member of V and its complement ~V is finite.

    I have proved that T is a topology on X, but I don't seem to be able to prove that;

    a) (X,T) is a Hausdorff space.
    b) (X,T) is a compact space.


    You don't say whether the sets V built as above are defined to be open or closed, but I'll assume they're open.

    Let x,y\in\left(X,T\right)\,,\,\,x\neq y . If both x\,,\,y\neq p then x\notin \{y\}\,,\,\,y\notin \{x\}\,,\,\,\{x\}\cap \{y\} = \emptyset and we're done, else: suppose x\neq y=p , then p\notin \{x\}\,,\,\,x\notin X\setminus \{x\} and

    again we're done since of course \{x\}\cap \left(X\setminus \{x\}\right)=\emptyset (Pay attention to the fact that in each case you MUST show the chosen subsets are open, each contains one

    of the points but not the other one and their intersection indeed is empty)

    Tonio
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