请输入您要查询的字词:

 

单词 ASpaceIsT1IfAndOnlyIfEverySubsetAIsTheIntersectionOfAllOpenSetsContainingA
释义

a space is T1 if and only if every subset A is the intersection of all open sets containing A


Say we have X, a T1-space, and A, a subset of X. We aim to show that the intersectionMathworldPlanetmath of all open sets containing A equals A. By de Morgan’s laws, that would be true if the complement of A, Ac, equalled the union of all closed setsPlanetmathPlanetmath in Ac. Let’s call this union of closed sets C.

Each set that makes up C is contained by Ac, so CAc. If we could show AcC, we’d be done.

Since X is T1, each singleton in Ac is closed (http://planetmath.org/ASpaceIsT1IfAndOnlyIfEverySingletonIsClosed). Their union, a subset of C, contains Ac, so we’re through.

Now suppose we know that in some topological spaceMathworldPlanetmath X, any subset A of X is the intersection of all open sets containing A. Given xy, we’re looking for an open set containing x but not y, to show that X is T1.

{x}=U openUxU

by hypothesisMathworldPlanetmathPlanetmath. If all open sets containing x contained y, y would be in the intersection; since y isn’t in the intersection, X must be T1.

随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 17:51:08