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.

Also see

• Results about the Sierpiński space can be found here.

Source of Name

This entry was named for Wacław Franciszek Sierpiński.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $11$. Sierpinski Space: $17$
• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $13 \text { - } 15$. Excluded Point Topology: $1$