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locally_finite_topology_subset [2016/09/14 15:52] nikolaj |
locally_finite_topology_subset [2016/09/15 01:42] nikolaj |
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| === 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). | * 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 $V$. | * Note that the name //locally compact// is already used for the situation where every point $x\in X$ has a compact neighborhood $V$. | ||
