Up till now we have focused almost entirely on the role of algebraic structures in mathematical physics. Occasionally, as in the previous chapter, it has been necessary to use some differential calculus, but this has not been done in any systematic way. Concepts such as continuity and differentiability, central to the area of mathematics known as analysis, are essentially geometrical in nature and require the use of topology for their rigorous definition. In broad terms, a topology is a structure imposed on a set to allow for the definition of convergence and limits of sequences or subsets. A space with a topology defined on it will be called a topological space, and a continuous map between topological spaces is one that essentially preserves limit points of subsets. The most general approach to this subject turns out to be through the concept of open sets.
Consider a two-dimensional surface embedded in Euclidean three-dimensional space . In this case we have an intuitive understanding of a ‘continuous deformation’ of the surface as being a transformation of the surface that does not involve any tearing or pasting. Topology deals basically with those properties that are invariant under continuous deformations of the surface. Metric properties are not essential to the concept of continuity, and since operations such as ‘stretching’ are permissible, topology is sometimes called ‘rubber sheet geometry’. In this chapter we will also define the concept of a metric space. Such a space always has a naturally defined topology associated with it, but the converse is not true in general – it is quite possible to define topology on a space without having a concept of distance defined on the space.
Contents
- 10.1 Euclidean topology
- 10.2 General topological spaces
- 10.3 Metric spaces
- 10.4 Induced topologies
- 10.5 Hausdorff spaces
- 10.6 Compact spaces
- 10.7 Connected spaces
- 10.8 Topological groups
- 10.9 Topological vector spaces
Concept index
Terms by section. Links lead to the brown underlined definitions in the Chinese reading text.
10.1 Euclidean topology
10.2 General topological spaces
- topology拓扑
- topological space拓扑空间
- standard topology标准拓扑
- closed set闭集
- relative topology相对拓扑
- subspace子空间
- accumulation point聚点
- cluster point聚点
- limit point极限点
- closure闭包
- interior内部
- boundary边界
- closed ball闭球
- dense set稠密集
- finer topology更细的拓扑
- stronger topology更强的拓扑
- coarser topology更粗的拓扑
- weaker topology更弱的拓扑
- indiscrete topology不可分拓扑
- discrete topology离散拓扑
- generated topology生成的拓扑
- open neighbourhood开邻域
- first countable第一可数
- second countable第二可数
- continuous map连续映射
- homeomorphism同胚
- homeomorphic同胚的
- topologically equivalent拓扑等价
- topological invariant拓扑不变量
10.3 Metric spaces
- metric space度量空间
- metric度量
- triangle inequality三角不等式
- metric topology度量拓扑
- Cauchy sequence柯西序列
- complete metric space完备度量空间
- normal space正规空间
10.4 Induced topologies
- induced topology诱导拓扑
- product topology积拓扑
- topological product拓扑积
- torus环面
- identification topology粘合拓扑
- real projective plane实射影平面
- real projective space实射影空间
10.5 Hausdorff spaces
10.6 Compact spaces
10.7 Connected spaces
10.8 Topological groups
- topological group拓扑群
- discrete group离散群
- topological subgroup拓扑子群
- closed subgroup闭子群
- component of the identity单位元分支
- locally connected局部连通
- open map开映射
- group of components分支群