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.