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 \(\mathscr{J}\) such that:

1. \(\emptyset\) and X \(\in\) \(\mathscr{J}\).
2. \(\mathscr{J}\) is closed under union of arbitrary sub-collection.
3. \(\mathscr{J}\) is closed under intersection of finite sub-collection.
   \(\mathscr{J}\) is called a collection of open sets, \((X,\mathscr{J})\) 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-\(\mathscr{J}\) 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, \(\mathscr{J}\) is the collection of V.

Def (Closure and Interior):Given set \(S\subset X\):
   Closure \(\overline{S}\)-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 \(\overline{A}\) 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 \(f:X \rightarrow Y\) 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\(\epsilon - \delta\) 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 \(\overline{M} = X\). 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."