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    properties of discrete spaces

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    I am looking at the properties of discrete spaces, particularly this one:

    Every discrete space is first-countable; it is moreover second-countable if and only if it is countable.

    How would this be proved?

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    Solution Preview

    This is just the matter of definitions.

    X is a discrete topoogical space if every subset of X is open. In particular, for each x in X we have {x} is an open set.

    X is first-countable, if for any x in X, there is a countable sequence {U_n(x), n>0} of open sets containing x, such that for any neighborhood V of x there exists a member U_i(x) of this sequence, such that U_i(x) is contained in V.
    Given a neighborhood V of x, we have that x is in V. But ...

    Solution Summary

    This posting offers help with calculating properties of discrete spaces.