Discrete space

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

Definition

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

• It has a basis comprising all the singleton subsets
• Every point is open
• Every subset is open
• Every subset is closed

Relation with other properties

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
totally disconnected space	the only connected subsets are singleton subsets
door space	the open subsets are precisely the same as the closed subsets
locally compact space
perfectly normal space
completely normal space
monotonically normal space
completely regular space
regular space
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.