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Math Help - Discrete Topology Question

  1. #1
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    Discrete Topology Question

    Prove that the topology T on X is the discrete Topology if and only if {x} \in T for all x \in X.

    I know what the definition for discrete topology is but im having a hard time prove this.
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  2. #2
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    Recall that a topology is closed under finite intersection and arbitrary union.

    If T is a topology on X having the property that \left( {\forall x \in X} \right)\left[ {\left\{ x \right\} \in T} \right]
    then if a\;\&\;b are two points in X what is \left\{ a \right\} \cap \left\{ b \right\}?

    What is \bigcup\limits_{x \in X} {\left\{ x \right\}}?

    If Y \subseteq X,\quad \bigcup\limits_{x \in Y} {\left\{ x \right\}}  = ?

    You said that you know the definition of discrete topology.
    What does the above show about T?
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