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导集.
在数学,特别是点集拓扑学中,拓扑空间的子集
的导集(导出集合)是
的所有极限点的集合。它通常记为
。
这个概念是格奥尔格·康托尔在1872年引入的,他开发集合论很大程度上就是为了研究在实直线上的导出集合。
导集是拓扑学的基础概念之一,可以用来定义拓扑空间。
给定集合
,考虑一个定义在
的幂集
上的运算
,若
满足以下导集公理,则称
为导集运算:
- D1:
![{\displaystyle d(\emptyset )=\emptyset }](https://wikimedia.org/api/rest_v1/media/math/render/svg/94424ad25632b19d1b7d03eb60c4ce9b2459125f)
- D2:
![{\displaystyle d(d(A))\subseteq d(A)\cup A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5df1c5d289971d5a9ee564a0692caad1868fae33)
- D3:
![{\displaystyle \forall x\in X,\ d(A)=d(A-\{x\})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97f5b034eb8c94fbbd8c9082b91d74704cd9598b)
- D4:
![{\displaystyle d(A\cup B)=d(A)\cup d(B)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50c13ec81a57fdae7c78605caebbb6329568dc2c)
称为
的导集。
从导集出发可以定义各种拓扑的基础概念:
- 闭集:
的子集
是闭集,当且仅当
。(从此处可以看到和闭集公理的等价性,从而可以等价地定义拓扑空间。)
- 同胚:拓扑空间
、
同胚,当且仅当存在双射
,使得
。
- 聚点
中的点称为
的聚点。
,若
,
,
。则称
和
是分离的。(注意:
不一定为
)。
- 集合
被定义为完美的,如果
。等价地说,完美集合是没有孤点的闭集。完美集合又称为完备集合。
- Cantor-Bendixson定理声称任何波兰空间都可以写为可数集合和完美集合的并集。因为任何波兰空间的
子集都再次是波兰空间,这个定理还证明了任何波兰空间的
子集都是可数集合和完美集合的并集。
- 拓扑空间
是T1 空间,当且仅当
。
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