## Locally finite topology subset

### Set

context | $\langle X,\mathcal T\rangle$ … topological space |

definiendum | ${\mathcal C} in it |

inclusion | ${\mathcal C}\subset \mathcal T$ |

for all | $x\in X$ |

exists | $V\in \mathcal T$ |

postulate | $x\in V$ |

postulate | $\{U\in {\mathcal C}\,|\,U\cap V\neq\emptyset\}$ … finite |

#### Idea

Like many properties, this is a notion of smallness. It's not about the smallness of a subset $U$ of $X$, but smallness of a collection ${\mathcal C}$ of subsets $U$ of $X$.

You may consider a well choosen sample of neighborhoods (the sets $V\in{\mathcal T}$) and ${\mathcal C}$ ought to be finite with respect to that sample (finite *pro* $V$).

#### Dicussion

- A topologal space is
*paracompact*if it any cover has a refinement with that property. - The sample of $V$'s above may be very big, so ${\mathcal C}$ is really only small w.r.t. the sample. In a
*compact*space, on the other hand, the cover itself is finite (and you don't need to consider that sample). - Note that the name
*locally compact*is already used for the situation where every point $x\in X$ has a compact neighborhood $V$.

#### Reference

Wikipedia: Locally finite collection, Paracompact space