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Math Help - The difference between and basis and subbasis for a topology

  1. #1
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    The difference between and basis and subbasis for a topology

    The two definitions seem very similar to me and is confusing.

    To express it simply

    a basis generates a topology T and a subbasis generates a basis for a topology?

    Is there an example that could clarify this difference?

    Thank you for your help.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: The difference between and basis and subbasis for a topology

    Quote Originally Posted by Jame View Post
    The two definitions seem very similar to me and is confusing.

    To express it simply

    a basis generates a topology T and a subbasis generates a basis for a topology?

    Is there an example that could clarify this difference?

    Thank you for your help.

    Roughly, bases are collections of subsets by which all open sets can be obatined via unions from that family and subbases are collection of open subsets for which any open subset can be obtained by unions of INTERSECTIONS of elements of the collection. For example the set of all infinite rectangles \{\mathbb{R}\times (a,b):a,b\in\mathbb{R}\} forms a subbasis for the usual topology on \mathbb{R}^2 but not a basis. Explain to me why.
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    Re: The difference between and basis and subbasis for a topology

    Oh i see. There are open sets in \mathBB{R}^2 which we cannot get from unioning these infinite rectangles together, but they can be made by unioning intersects of these rectangles.

    Thank you for thee clarification. That is a really good example.
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