Discrete space

Definition
A discrete space is a topological space satisfying the following equivalent conditions:


 * 1) It has a basis of open subsets comprising all the singleton subsets
 * 2) Every singleton subset is an open subset
 * 3) Every subset is an open subset
 * 4) Every subset is a closed subset
 * 5) Every subset is a clopen subset

Given any set, there is a unique topology on it making it into discrete space. This is termed the discrete topology. The discrete topology on a set is the finest possible topology on the set.

Related properties
Compactness is the opposite of discreteness in some sense. The only topological spaces that are both discrete and compact are the finite spaces.

Metaproperties
A (finite?) direct product of discrete spaces is discrete.

Any subspace of a discrete space is discrete under the induced topology.