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单词 ExampleOfAnAlexandroffSpaceWhichCannotBeTurnedIntoATopologicalGroup
释义

example of an Alexandroff space which cannot be turned into a topological group


Let denote the set of real numbers and τ={[a,)|a}{(b,)|b}. One can easily verify that (,τ) is an Alexandroff space.

PropositionPlanetmathPlanetmathPlanetmath. The Alexandroff space (,τ) cannot be turned into a topological groupMathworldPlanetmath.

Proof. Assume that =(,τ,) is a topological group. It is well known that this implies that there is H which is open, normal subgroupMathworldPlanetmath of . This subgroupMathworldPlanetmathPlanetmath ,,generates” the topologyPlanetmathPlanetmath (see the parent object for more details). Thus H because τ is not antidiscrete. Let g such that gH (and thus gHH=). Then gH is again open (because the mapping f(x)=gx is a homeomorphism). But since both H and gH are open, then gHH. Indeed, every two open subsets in τ have nonempty intersectionMathworldPlanetmath. ContradictionMathworldPlanetmathPlanetmath, because diffrent cosets are disjoint.

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更新时间:2025/5/4 5:56:00