# Definition:Sierpiński Space

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## 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: $1$