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Topological Space
- Topology 1.1

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


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