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Topology: Indiscrete/Discrete Topology

  1. Sep 5, 2009 #1
    I am reading from my text, and was just wondering if someone could provide additional information on the following examples.

    0.1 Examples. For any set X each of the following defines a topology for X.

    (1) [tex]T_{*} = {A \subseteq X|a \in A \Rightarrow X \subseteq A},[/tex] Indiscrete Topology.

    (2) [tex]T^{*} = {A \subseteq X|a \in A \Rightarrow {a} \subseteq A},[/tex] Discrete Topology.

    Questions:
    I was wondering how we can have the following statement (from above),
    (1) [tex]a \in A \Rightarrow X \subseteq A[/tex]
    (2) [tex]a \in A \Rightarrow {a} \subseteq A[/tex]

    Thanks,


    JL
     
  2. jcsd
  3. Sep 6, 2009 #2
    This is a really set-theoretic definition approach to defining these topologies, and they threw me off at first too. The indiscrete topology of X is the topology containing only the empty set and X itself. This agrees with their definition because if A=empty set, then nothing's in there so the implication to the right of the bar is "true" (I think some might say vacuously true, but I don't like the term). So A=empty set is in the topology. If X=A, then X is automatically a subset of A and thus in the topology. These are the only two subsets of X that satisfy their definition.

    The discrete topology of X is just the collection of all subsets of X, i.e. the topology equals the power set of X. This again agrees with their definition because any subset A of X will satisfy the property that if [itex]a\in A[/itex], then [itex]a\subseteq A[/itex].

    I think defining these topologies in this way and not explaining them is very poor writing. I don't see any reason why they would do so, because the definitions I gave (which was what I was taught and is in the book Topology by Munkres) are perfectly rigorous. Their definitions certainly give no immediate insight as to what the topologies actually consist of.
     
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