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    Regular topological spaces

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    A) Reals with the "finite complement topology:" U open in X if U - X is finite or is all of X. Why compact?
    Why not regular?
    b) Reals with the "countable complement topology:" U open in X if X - U is countable or is all of X. Why not
    compact? Why not regular.
    c) Reals with the "K-topology:" basis consists of open intervals (a,b) and sets of form (a,b) - K where K = {1, 1/2, 1/3, ... } Why not compact? Why not regular?

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    Solution Preview

    A topological space X is compact if and only any open cover of X has a finite subcover.
    A topological space is regular if and only if it satisfies the T3 axiom: For any point x and closed set A which doesn't contain x, there exists open neighborhood U(x) and V(A) such that U(x) intersects V(A) is empty.
    We use X to denote the space of reals.
    a) X is compact. Suppose C is an open cover of X. We select an open set B in C. By the defintion, X-B is finite. Suppose X-B has n points, say b1,b2,...,bn. So bi is not in B for any 1<=i<=n. For any bi, we can find an open set Ci in C such that bi is in Ci. If not, then bi doesn't belong to any open set in C and thus bi is not in the union of all open sets in C and thus bi is not in X, this is a contradiction. Therefore, {B,C1,C2,...,Cn} is ...

    Solution Summary

    This explains statements regarding compact and regular sets.