Discrete space

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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:

  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.

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.