# One-point space

From Topospaces

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:

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