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:
- \(\emptyset\) and X \(\in\) \(\mathscr{J}\).
- \(\mathscr{J}\) is closed under union of countable sub-collection.
- \(\mathscr{J}\) 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 \(\overline{S}\)-collection of all adherent points in \(S\),
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 \(A\).
由上文可知,closure的另一个等价定义是所有包含S的closed set的交集。
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."
标签:set,mathscr,拓扑,前置,泛函,collection,closed,open,Def
From: https://www.cnblogs.com/rzdclp/p/18403001