拓扑性质之紧致性总结

几种紧的描述方式

1.列紧

一个拓扑空间(X,τ)(X,\tau)(X,τ)称之为列紧的，如果它的每个序列有收敛的子序列。

ps：这里的收敛指的是序列的收敛，而不是集合的收敛，两种收敛存在区别。

2.子集紧

一个拓扑空间(X,τ)(X,\tau)(X,τ)称之为子集紧的，如果XXX的每个无穷子集都有XXX 中极限点。

3.可数紧

一个拓扑空间(X,τ)(X,\tau)(X,τ)称之为可数紧的，如果其任何可数开覆盖都有有限子覆盖。

4.紧致

一个拓扑空间(X,τ)(X,\tau)(X,τ)称为紧致的，如果它的任何开覆盖存在有限子覆盖。

ps：覆盖的定义即为子集族。

5.伪紧

若拓扑空间(X,τ)(X,\tau)(X,τ)满足在连续函数下的像是有界的，则称XXX是伪紧的。

可以看到，五种对紧的定义是从三个角度出发对拓扑空间进行的刻画：

1.子集紧、可数紧是从极限点角度出发对拓扑空间进行刻画。

2.可数紧、紧致则是从覆盖数量角度出发对拓扑空间进行刻画。

3.伪紧从连续函数像的有界性对拓扑空间进行刻画。

ps：“有界”这个定义是在距离空间中的，如果只给定了拓扑空间是不够的。

那么随之而来的Question就是：

“紧”到底是一种什么性质，为什么它的名字叫“紧”？

个人理解：
紧就是挤得慌，东西多，地方小。比如压缩饼干，折叠起来的降落伞，一辆超载的面包车。

为什么极限点，覆盖能够表述紧这个性质呢？

我们假如放出了ℵ1\aleph_1ℵ1个罪犯，让他们去城市中游荡，那我就得做好派出至多ℵ1\aleph_1ℵ1个武警去抓捕他们的准备。但是如果让这些罪犯一块乘坐可数辆公交车(超载危险，请勿模仿)，那我派出可数个警察就可以了。这就是我对覆盖的理解。

同样是那被放出的ℵ1\aleph_1ℵ1个罪犯，如果让他们分散跑，那一个罪犯周围很难找到别的罪犯，如果让他们挤在公交车上，那一个罪犯周围必然能找到别的罪犯。

把上面的原因结果反过来想。结果就是对原因性质的一种表述，这种原因就是紧。

我们下面来总结下：

各种紧性质间的关系

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Lindelof

列紧

可数紧

伪紧

子集紧

紧性