One-point space

This article is about a particular topological space (uniquely determined up to homeomorphism)|View a complete list of particular topological spaces

Definition

The one-point space is a topological space whose underlying set has exactly one point. There is a unique topology on any space of one point: all subsets must be open as well as closed. The one-point space can also be defined in the following equivalent ways:

1. It is the terminal object in the category of topological spaces.
2. It is a nonempty space equipped with a topology that is both the discrete topology and the trivial topology.

Relation with other properties

Weaker properties

• Finite space
• Metrizable space
• Compact Hausdorff space
• Hausdorff space
• Compact space