discrete space
The discrete topology on a set $X$ is the topology^{} given by the power set^{} of $X$. That is, every subset of $X$ is open in the discrete topology. A space equipped with the discrete topology is called a discrete space.
The discrete topology is the http://planetmath.org/node/3290finest topology one can give to a set. Any set with the discrete topology is metrizable by defining $d(x,y)=1$ for any $x,y\in X$ with $x\ne y$, and $d(x,x)=0$ for any $x\in X$.
The following conditions are equivalent^{}:

1.
$X$ is a discrete space.
 2.

3.
Every subset of $X$ containing $x$ is a neighborhood^{} of $x$.
Note that any bijection between discrete spaces is trivially a homeomorphism^{}.
Discrete Subspaces
If $Y$ is a subset of $X$, and the subspace topology on $Y$ is discrete, then $Y$ is called a discrete subspace or discrete subset of $X$.
Suppose $X$ is a topological space and $Y$ is a subset of $X$. Then $Y$ is a discrete subspace if and only if, for any $y\in Y$, there is an open $S\subset X$ such that
$$S\cap Y=\{y\}.$$ 
Examples

1.
$\mathbb{Z}$, as a metric space with the standard distance metric $d(m,n)=mn$, has the discrete topology.

2.
$\mathbb{Z}$, as a subspace^{} of $\mathbb{R}$ or $\u2102$ with the usual topology, is discrete. But $\mathbb{Z}$, as a subspace of $\mathbb{R}$ or $\u2102$ with the trivial topology, is not discrete.

3.
$\mathbb{Q}$, as a subspace of $\mathbb{R}$ with the usual topology, is not discrete: any open set containing $q\in \mathbb{Q}$ contains the intersection^{} $U=B(q,\u03f5)\cap \mathbb{Q}$ of an open ball around $q$ with the rationals. By the Archimedean property, there’s a rational number^{} between $q$ and $q+\u03f5$ in $U$. So $U$ can’t contain just $q$: singletons can’t be open.

4.
The set of unit fractions $F=\{1/n\mid n\in \mathbb{N}\}$, as a subspace of $\mathbb{R}$ with the usual topology, is discrete. But $F\cup \{0\}$ is not, since any open set containing $0$ contains some unit fraction.

5.
The product^{} of two discrete spaces is discrete under the product topology. The product of an infinite^{} number of discrete spaces is discrete under the box topology, but if an infinite number of the spaces have more than one element^{}, it is not discrete under the product topology.
Title  discrete space 
Canonical name  DiscreteSpace 
Date of creation  20130322 12:29:56 
Last modified on  20130322 12:29:56 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  17 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 5400 
Synonym  discrete topological space 
Related topic  Discrete2 
Defines  discrete subspace 
Defines  discrete topology 
Defines  discrete space 
Defines  discrete subset 