# irreducible

A subset $F$ of a topological space^{} $X$ is reducible if it can be written as a union $F={F}_{1}\cup {F}_{2}$ of two closed proper subsets^{} ${F}_{1}$, ${F}_{2}$ of $F$ (closed in the subspace topology). That is, $F$ is reducible if it can be written as a union $F=({G}_{1}\cap F)\cup ({G}_{2}\cap F)$ where ${G}_{1}$,${G}_{2}$ are closed subsets of $X$, neither of which contains $F$.

A subset of a topological space is irreducible (or *hyperconnected*) if it is not reducible.

As an example, consider $\{(x,y)\in {\mathbb{R}}^{2}:xy=0\}$ with the subspace topology from ${\mathbb{R}}^{2}$. This space is a union of two lines $\{(x,y)\in {\mathbb{R}}^{2}:x=0\}$ and $\{(x,y)\in {\mathbb{R}}^{2}:y=0\}$, which are proper closed subsets. So this space is reducible, and thus not irreducible.

Title | irreducible |
---|---|

Canonical name | Irreducible1 |

Date of creation | 2013-03-22 12:03:30 |

Last modified on | 2013-03-22 12:03:30 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 14 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 14A15 |

Classification | msc 14A10 |

Classification | msc 54B05 |

Related topic | IrreducibleComponent |

Related topic | HyperconnectedSpace |

Defines | reducible |