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Math Help - Empty Set as a union of Basis Elements

  1. #1
    Senior Member slevvio's Avatar
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    Empty Set as a union of Basis Elements

    I just had a quick question.

    If \mathcal{B} is a basis for a topology on a set X, defined in the way

    1)  X = \displaystyle\bigcup_{B \in \mathcal{B}} B
    2) If B_1, B_2 \in \mathcal{B}, and  x \in B_1 \cap B_2, then there exists  B_3 \in \mathcal{B} such that x \in B_3 \subseteq B_1 \cap B_2

    Then it is a fact that this generates a topology on X, where the sets in the topology are unions of elements of \mathcal{B}. But we require \emptyset to be an element in this topology - is it ok to take a union of none of the basis elements?

    Thanks for any help.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Empty Set as a union of Basis Elements

    By definition \big\cup\{B_i:i\in\emptyset\}=\emptyset .
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