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If I am correct, a discrete subset of a topological space is defined to be a subset consisting of isolated points only. This is actually equivalent to that the subspace topology on the subset is discrete topology. There seems no restriction on the cardinality of a discrete subset, i.e. its cardinality can be any.

  1. I was wondering if the following quote from wolfram is true and why?

    Typically, a discrete set is either finite or countably infinite.

    What kinds of topological spaces are "typical"?

  2. Added: Is the following quote from the same link true

    On any reasonable space, a finite set is discrete.

    What kinds of topological spaces does "reasonable" mean?

  3. Is discrete mathematics always under the setting of discrete sets wrt some topologies? In other words, is it a special case of topology theory? Or can it exist without topology?

Thanks and regards!

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