# Sierpinski space

*Sierpinski space* is the topological space^{} $X=\{x,y\}$ with the topology given by $\{X,\{x\},\mathrm{\varnothing}\}$.

Sierpinski space is ${T}_{0}$ (http://planetmath.org/T0) but not ${T}_{1}$ (http://planetmath.org/T1). It is ${T}_{0}$ because $\{x\}$ is the open set containing $x$ but not $y$. It is not ${T}_{1}$ because every open set $U$ containing $y$ (namely $X$) contains $x$ (in other words, there is no open set containing $y$ but not containing $x$).

Remark. From the Sierpinski space, one can construct many non-${T}_{1}$ ${T}_{0}$ spaces, simply by taking any set $X$ with at least two elements, and take any non-empty proper subset^{} $U\subset X$, and set the topology $\mathcal{T}$ on $X$ by $\mathcal{T}=P(U)\cup \{X\}$.

Title | Sierpinski space |
---|---|

Canonical name | SierpinskiSpace |

Date of creation | 2013-03-22 12:06:26 |

Last modified on | 2013-03-22 12:06:26 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 9 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 54G20 |

Synonym | Sierpiński space |

Related topic | T1Space |

Related topic | T2Space |

Related topic | SeparationAxioms |