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Math Help - possible errata

  1. #1
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    possible errata

    this is part of a theorem in my topology book:

    A family B of subsets of a set X is a base for a topology of X if and only if B has the following two properties:
    1. each x in X lies in at least one set in B
    2. if U,V are in B and x is in the intersection of X and V, then there exists W in B such that x is in W and W is a subset of U intersect V.

    i was wondering, for #1, shouldnt each x be in T where T is a topology for X? (note: the book states that a base has property 1 because X is open)
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  2. #2
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    Although this is not a standard definition of a basis for a topology, I think that your text is correct in the way it stated #1. Any topology on X includes X, so X is by definition an open set.
    As I see the statement of #1, it insures that B "covers" all points in X.
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  3. #3
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    ah i see, thanks. and what would we need to show to verify that base of topology satisfies condition 2? is it basically saying that for all x in U intersect V, theres a smaller union of the elements in B that is contained in the whole of U intersect V?
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