# 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 $x\neq y$, and $d(x,x)=0$ for any $x\in X$.

1. 1.

$X$ is a discrete space.

2. 2.

Every singleton in $X$ is an open set.

3. 3.

## 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. 1.

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

2. 2.

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

3. 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  $U=B(q,\epsilon)\cap\mathbb{Q}$ of an open ball around $q$ with the rationals. By the Archimedean property, there’s a rational number  between $q$ and $q+\epsilon$ in $U$. So $U$ can’t contain just $q$: singletons can’t be open.

4. 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. 5.
 Title discrete space Canonical name DiscreteSpace Date of creation 2013-03-22 12:29:56 Last modified on 2013-03-22 12:29:56 Owner mathcam (2727) Last modified by mathcam (2727) Numerical id 17 Author mathcam (2727) Entry type Definition Classification msc 54-00 Synonym discrete topological space Related topic Discrete2 Defines discrete subspace Defines discrete topology Defines discrete space Defines discrete subset