泛函分析-0：前置准备——拓扑空间

拓扑空间是比度量空间更一般的定义，或者说，度量空间是在拓扑空间中引入了关于‘距离’的概念，它比一般拓扑空间更侧重于不同的‘距离’测量方法。
Def (Topology): given set X, a topology is a collection of X's subsets $\mathcal{J}$ such that:

1. $\mathrm{\varnothing }$ and X $\in$ $\mathcal{J}$.
2. $\mathcal{J}$ is closed under union of arbitrary sub-collection.
3. $\mathcal{J}$ is closed under intersection of finite sub-collection.
   $\mathcal{J}$ is called a collection of open sets, $(X,\mathcal{J})$ is called a topological space.

这事实上让我想到了$\sigma -algebra$的定义，两者有很大的相似之处，不过这不是这篇的主题。
下面是一些关于拓扑空间的例子。
Example 1: Discrete Topology-$\mathcal{J}$ is the power set of $X$.
离散空间是一个非常完美的空间，因为你不可能再将它细分。观察到离散空间是由离散度量引出的——离散度量将每个点都‘分割’开，使得在这个空间中的每一个点都存在于一个独立的open ball(unit ball)中，因此任意序列都不可能无限接近于任意点，讨论收敛的唯一可能就是常量序列。

Example 2: Zariski Topology-k is an infinite field, V$\subset$k is closed iff V is finite, $\mathcal{J}$ is the collection of V.

Def (Closure and Interior):Given set $S\subset X$:
   Closure $\bar{S}$-intersection of all closed sets containing $S$,
   Interior $S^o$-union of all open balls within $S$.

Remark1: The closure of a closed ball may NOT be its closed ball (under some metric).

Example 3: Consider the discrete metric under $X$ and a unit ball $B$, then its closure is it self, but its closed set contains every element in $X$.

Remark2: The closure $\bar{A}$ is the smallest closed set containing $A$, while the interior $A^o$ is the largest open set contained in $A$.

Def (Neighborhood): For $x\in X$, $N\subset X$ is the neighborhood of $x$ if $x\in N^o$.

接下来是本节最重要的概念。

Def (Continuity): A map $f:X\to Y$ is continuous iff the preimage of open sets are open.

这是对于连续函数的（拓扑风格）定义，对于函数上一个点的连续性，我们也可以采用以上方法去定义（而不是使用$ϵ-\delta$ 语言)。

Def (Point Continuity): f is said to be continuous at some point $x$ if $V$ is an open neighborhood of $f(x)$, then the preimage of $V$ contains an open neighborhood U of $x$.

接下来是本篇最后一个概念。

Def (Separability): $X$ is a topological space and $f$ is a map, we say $M\subset X$ is dense if $\bar{M}=X$. We say $X$ is separable if $X$ has countable dense set.

关于这个定义，我想在此摘抄一段我从stackexchange上看到的评论：
"My understanding is it comes from the special case of R, where it means that any two real numbers can be separated by, say, a rational number."