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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 (countable pro $V$).

Dicussion

A topologal space is paracompact if it has a cover with that property.

Reference

Subset of

Context

Requirements*

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