===== Locally finite topology subset =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $\langle X,\mathcal T\rangle$ ... topological space |
| @#FFBB00: definiendum | @#FFBB00: ${\mathcal C} in it |
| @#AAFFAA: inclusion   | @#AAFFAA: ${\mathcal C}\subset \mathcal T$ |
| @#FFFDDD: for all     | @#FFFDDD: $x\in X$ |
| @#FFFDDD: exists      | @#FFFDDD: $V\in \mathcal T$ |
| @#55EE55: postulate   | @#55EE55: $x\in V$ |
| @#55EE55: postulate   | @#55EE55: $\{U\in {\mathcal C}\,|\,U\cap V\neq\emptyset\}$ ... finite |

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=== 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:
[[https://en.wikipedia.org/wiki/Locally_finite_collection | Locally finite collection]],
[[https://en.wikipedia.org/wiki/Paracompact_space | Paracompact space]]

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=== Subset of ===
[[Cover]]
=== Context ===
[[Topological space]]
=== Requirements* ===
[[Cover]]