外部
Multi tool use
在拓扑学中，拓扑空间 X 一集合 S 的外部，是所有不交于 S 的开集之并集。它自身是开集。S 的外部可記为 ext S 或 Se。S 的外點是 S 的外部的元素。
等价定义
S 的外部等价于 S 的闭包的补集，也等價於 S 在 X 中的补集的内部。
性質
許多外部的性質可直接從內部得出，例如
- ext(S)是不交於S的開集。
- ext(S)是所有不交於S的開集之聯集。
- ext(S)是不交於S之開集中最大的。
- 如果S是T的子集，則ext(S)是ext(T)的父集。
不像內部算子，外部算子並非冪等，但有如下性質
点集拓扑系列
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| 基本概念 |
连续函数 · 同胚 · 子空間 · 積空間 · 商空間 · 序空間
邻域 · 內部 · 邊界 · 外部 · 極限點 · 孤点
基 · 鄰域系統 · 开集 · 闭集 · 闭开集 · 稠密集 · 无处稠密集 · 闭包
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| 拓扑空间 |
實數線 · 离散空间 · 密着拓扑 · 余有限空间 · 下限拓扑 · 康托尔集
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| 连通空间 |
连通空间 · 局部连通空间 · 道路连通空间 · 单连通 · N-连通 · 不可約空間
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| 紧空间 |
可数紧 · 序列紧 · 聚点紧 · 局部紧
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| 一致空间 |
一致同构 · 一致性质 · 一致收敛 · 一致连续
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| 可數性公理 |
第一可數 · 第二可數 · 可分空间 · 林德勒夫空間
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| 分离公理 |
柯尔莫果洛夫空间 · T1空间 · 豪斯多夫空间 · 正则空间 · 吉洪诺夫空间 · 正规空间
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| 定理 |
- 波尔查诺-魏尔斯特拉斯定理
- 海涅-博雷尔定理
- 贝尔纲定理
- 吉洪诺夫定理
- 乌雷松引理
- 乌雷松度量化定理
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