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单词 UniversalNetsInCompactSpacesAreConvergent
释义

universal nets in compact spaces are convergent


Theorem - A universal net (xα)α𝒜 in a compact space X is convergent.

Proof : Suppose by contradictionMathworldPlanetmathPlanetmath that (xα)α𝒜 was not convergent. Then for every xX we would find neighborhoodsMathworldPlanetmathPlanetmath Ux such that

α𝒜αα0xα0Ux

The collectionMathworldPlanetmath of all this neighborhoods cover X, and as X is compactPlanetmathPlanetmath, a finite numberUx1,Ux2,,Uxn also cover X.

The net (xα)α𝒜 is not eventually in Uxk so it must be eventually in X-Uxk (because it is a net). Therefore we can find αk𝒜 such that

αkαxαX-Uxk

Because we have a finite number α1,α2,αn𝒜 we can find γ𝒜 such that αkγ for each 1kn.

Then xγX-Uxk for all k, i.e. xγUxk for all k. But Ux1,Ux2,,Uxn cover X and thus we have a contradiction.

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更新时间:2025/5/4 22:08:13