Definition:Sierpiński Space

Definition
The Sierpiński space is a particular point space with exactly two elements.

Its usual presentation is:
 * $T = \left({\left\{{0, 1}\right\}, \left\{{\varnothing, \left\{{0}\right\}, \left\{{0, 1}\right\}}\right\}}\right)$

that is, as a particular point topology on the set $\left\{{0, 1}\right\}$ where the particular point is $0$.

It can also immediately be seen to be an excluded point topology on the set $\left\{{0, 1}\right\}$ where the excluded point is $1$.

The Sierpiński space is considered to be a trivial instance of both the particular point topology and the excluded point topology.