\begin{definition}[Definition:Sierpiński Space]
The '''Sierpiński space''' is a particular point space with exactly two elements.
Its usual presentation is:
:$T = \struct {\set {0, 1}, \set {\O, \set 0, \set {0, 1} } }$
that is, as a particular point topology on the set $\set {0, 1}$ where the particular point is $0$.
It can also immediately be seen to be an excluded point topology on the set $\set {0, 1}$ 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.
\end{definition}