One-point space

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.

Weaker properties

 * Stronger than::Finite space
 * Stronger than::Metrizable space
 * Stronger than::Compact Hausdorff space
 * Stronger than::Hausdorff space
 * Stronger than::Compact space