# Sierpiński space

From Topospaces

This article is about a particular topological space (uniquely determined up to homeomorphism)|View a complete list of particular topological spaces

## Contents

## Definition

### Explicit definition

The **Sierpiński space** is a topological space defined as follows (up to homeomorphism):

- The underlying set is a two-point set .
- The open subsets are: . Thus, the closed subsets are .

### Definition as a left order topology

The Sierpiński space can be defined as the topological space arising by taking the left order topology on a totally ordered set of size two.

## Topological space properties

Property | Satisfied? | Explanation | Corollary properties satisfied/dissatisfied |
---|---|---|---|

Separation | |||

Kolmogorov space (the axiom) | Yes | ||

T1 space | No | The subset is not closed. | dissatisfies: Hausdorff space |

regular space | No | Consider the point and the closed subset . These cannot be separated by disjoint open subsets. | |

normal space | Yes | follows from being ultraconnected | |

Cardinality | |||

finite space | Yes | satisfies: compact space and all corollaries thereof satisfies: second-countable space and all corollaries thereof | |

Connectedness | |||

connected space | Yes | ||

path-connected space | Yes | the function that sends all elements to except 1 which is sent to is a continuous function. | satisfies: connected space |

irreducible space | Yes | the only proper non-empty closed subset is , so the space cannot be expressed as a union of two such subsets. | satisfies: connected space |

ultraconnected space | Yes | the only proper non-empty closed subset is , so the condition is vacuously satisfied. | satisfies: path-connected space, connected space, normal space |

locally path-connected space | Yes | satisfies: locally connected space | |

Homotopy-invariant properties | |||

contractible space | Yes | satisfies: weakly contractible space, simply connected space, acyclic space | |

Discreteness | |||

discrete space | No | ||

door space | Yes | satisfies: submaximal space, irresolvable space, hereditarily irresolvable space |