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单词 AlexandroffSpaceIsT1IfAndOnlyIfItIsDiscrete
释义

Alexandroff space is T1 if and only if it is discrete


PropositionPlanetmathPlanetmathPlanetmath. Let X be an Alexandroff space. Then X is T1 if and only if X is discrete.

Proof. ,,” It is easy to see, that every discrete space is Alexandroff and T1.

,,” Recall that topological spaceMathworldPlanetmath is T1 if and only if every subset is equal to the intersectionMathworldPlanetmath of all its open neighbourhoods. So let xX. Then the intersection of all open neighbourhoods {x}o of x is equal to {x}. But since X is Alexandroff, then {x}o={x} is open and thus points are open. Therefore X is discrete.

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更新时间:2025/5/4 15:22:47