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
Subset of
Context
Requirements*
