Definition:Sierpiński Space

Definition
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.

Also known as
Many sources render the name without the diacritic on the n, that is, as Sierpinski space.

This eases presentation in media in which it is inconvenient to render non-ASCII characters.