# properties of discrete spaces

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?

© BrainMass Inc. brainmass.com October 10, 2019, 2:25 am ad1c9bdddfhttps://brainmass.com/math/geometry-and-topology/properties-discrete-spaces-376641

#### 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.

$2.19