# Discrete space

From Topospaces

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

*This is an opposite of compactness*

## Contents

## Definition

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

- It has a basis of open subsets comprising all the singleton subsets
- Every singleton subset is an open subset
- Every subset is an open subset
- Every subset is a closed subset
- 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.

## Relation with other properties

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Disconnectedness type | ||||

totally disconnected space | the only connected subsets are singleton subsets | |||

door space | every subset is open or closed | |||

submaximal space | every subset is locally closed | |||

weakly submaximal space | every finite subset is locally closed | |||

zero-dimensional space | has a basis of clopen subsets | |||

Alexandrov space | arbitrary intersection of open subsets is open | |||

almost discrete space | Alexandrov and zero-dimensional | |||

extremally disconnected space | every regular open subset is closed | |||

Separation type | ||||

locally compact space | ||||

perfectly normal space | ||||

completely normal space | ||||

monotonically normal space | ||||

completely regular space | ||||

regular space | ||||

Extra structure type | ||||

metrizable space | ||||

CW-space | ||||

polyhedron |

### 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

### Products

This property of topological spaces is closed under taking arbitrary products

View all properties of topological spaces closed under products

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

### Hereditariness

This property of topological spaces is hereditary, or subspace-closed. In other words, any subspace (subset with the subspace topology) of a topological space with this property also has this property.

View other subspace-hereditary properties of topological spaces

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