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locally_finite_topology_subset [2016/09/14 15:34] nikolaj |
locally_finite_topology_subset [2016/09/15 01:42] (current) nikolaj |
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| === Idea === | === 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 $X$. | + | 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 collection of neighborhoods (the sets $V\in{\mathcal T}$) and ${\mathcal C}$ ought to be countable w.r.t. to that sample, i.e. pro $V$. | + | 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 === | === Dicussion === | ||
| - | A topologal space is paracompact if it has a cover with that property. | + | * 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 === | === Reference === | ||
