学习笔记《Topology》

拓扑学，最基础的理解就是「七桥问题」和「四色地图问题」，我也是在这个基础上深入这个领域进行初步学习的

不清楚为什么学习 Topology 的时候，内容却大部分是集合论的东西，这个等以后学习集合论的时候，再回头修订这块的内容~

Topology 的视频介绍：

https://www.youtube.com/watch?v=AmgkSdhK4K8
https://www.youtube.com/watch?v=FhSFkLhDANA

文字介绍：

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, that satisfy certain properties, turning the given set into what is known as a topological space. Important topological properties include connectedness and compactness.

Topology developed as a field of study out of geometry and set theory, through analysis of concepts such as space, dimension, and transformation. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs (Greek-Latin for "geometry of place") and analysis situs (Greek-Latin for "picking apart of place"). Leonhard Euler's Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics.

偏理论的一个课程地址：
https://www.youtube.com/watch?v=FHL4udeLf9Q

Neighborhoods

用来描述某个点周围的点的集合

d(a,b) 或者 | a - b | 表示两个点之间的距离

Open sets and Closed sets

Open sets 用来描述一个不包含「端点」的集合：

Closed sets 用来描述一个包含了「端点」的集合：

一个集合可以既不是 Open 的，也不是 Closed 的：

一个集合可以同时是 Open 和 Closed 的，叫做 Clopen 集合：

interior point, exterior point and boundary point

这样更直观的理解：

• 没有 buoundary point 的 set 就是 clopen set

Closure

Disconnect and Totally disconnect

Cluster points

这个后面会再单独详细学习

Cantor set

这个后面会再单独详细学习