Topological Space
- Topology 1.1
from 「Topology」: James R. Munkres
Topological space
Topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness.
Given a set $X$, a set $\mathcal{T}\sub\mathcal{P}(X)$ is called the topology if it satifies following conditions. If then, we call the pair $(X,\mathcal{T})$ a topological space.
- $ \empty,X \in \mathcal{T} $
- $ S\sube\mathcal{T} \Rightarrow \bigcup S\sube\mathcal{T} $
- $ U,V\in\mathcal{T} \Rightarrow U\cap V\in\mathcal{T} $
Every set $X$ has its trivial topology $\set{\empty,X}$ and discrete topology $\mathcal{P}(X)$.
Open and closed set
We usually define an open set as an element of a topolofical space, and closed set the complement of them.
For a topological space $(X,\mathcal{T})$,
- $ U\in\mathcal{T} \Rightarrow U: \text{open set} $
- $ F\sub X \;\text{s.t.}\; X\setminus F\in\mathcal{T} \Rightarrow F: \text{closed set} $
- If a set is both open and closed, it is called clopen set.
Examples
- Hausdorff spaces
- Hilbert spaces
- Metric spaces
- Proximity spaces
- Uniform spaces
- Function spaces
- Cauchy spaces
