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Discrete topology?

  1. Jun 10, 2007 #1
    Defn: the discrete topology on X is defined by letting the topology consist of all open subsets of X.

    Why do they use the word discrete in the term discrete topology? Is it because there are subsets such that each subset contain only one point in the space. And these collection of subsets are in the topology so its called the discrete topology as each point is contained in a subset which is in the so called discrete topology.

    How do you usually show a set has the discrete topology?
  2. jcsd
  3. Jun 10, 2007 #2

    matt grime

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    Haven't we answered this question recently? In the discrete topology every point lies in an open set that intersects no other points. Discrete means 'separated', like the integers, say - there is some 'space' between them. In topology, the only notion of 'space' is separatedness.

    I can't say I've ever seen anything other than trivial questions asking you to show that some topology is equivalent to the discrete topology, but you'd just have to show that points are open sets.
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