# closed point

Let $X$ be a topological space^{} and suppose that $x\in X$. If $\{x\}=\overline{\{x\}}$ then we say that $x$ is a
*closed point*. In other words, $x$ is closed if $\{x\}$ is a closed set.

For example, the real line $\mathbb{R}$ equipped with the usual metric topology^{}, every point is a closed point.

More generally, if a topological space is ${T}_{1}$ (http://planetmath.org/T1), then every point in it is closed. If we remove the condition of being ${T}_{1}$, then the property fails, as in the case of the Sierpinski space $X=\{x,y\}$, whose open sets are $\mathrm{\varnothing}$, $X$, and $\{x\}$. The closure^{} of $\{x\}$ is $X$, not $\{x\}$.

Title | closed point |
---|---|

Canonical name | ClosedPoint |

Date of creation | 2013-03-22 16:22:24 |

Last modified on | 2013-03-22 16:22:24 |

Owner | jocaps (12118) |

Last modified by | jocaps (12118) |

Numerical id | 9 |

Author | jocaps (12118) |

Entry type | Definition |

Classification | msc 54A05 |