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Locally finite topology subset
Set
| context | $\langle X,\mathcal T\rangle$ … topological space |
| definiendum | $C$ in it |
| inclusion | $C\subset \mathcal T$ |
| for all | $x\in X$ |
| exists | $V\in \mathcal T$ |
| postulate | $x\in V$ |
| postulate | $\{U\in C\,|\,U\cap V\neq\emptyset\}$ … finite |
Idea
Like many properties, this is a notion of smallness, and more particularly, not smallness of a subset of $X$ but smallness of a collection $C$ of subsets of $X$.
You choose a small sample of neighborhoods (the sets $V\in\mathcal T$) and $C$ ought to be countable w.r.t. to that sample, i.e. pro $V$.
Dicussion
A topologal space is paracompact if it has a cover with that property.
Reference
Wikipedia: Locally finite collection, Paracompact space
Subset of
Context
Requirements*
