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 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.