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所在平台: Udemy |
课程主页: https://www.udemy.com/course/topology/
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课程名称：拓扑空间入门 概述：该课程旨在探索如何在给定的点集$X$中精确定义“接近性”和“局部性”。虽然我们可以定义个别点之间的距离，从而推导出相应的概念，但更微妙的方法是为这个集合定义一个称为“拓扑”的结构，使得$X$成为一个“拓扑空间”，从而使得接近性、局部性以及连续性（保持接近性的性质）在$X$中得以精确定义。课程将深入探讨与拓扑空间相关的连接性（以及不连接性）、紧致性和极限等概念。 课程内容主要基于Munkres的第二章及其练习，但通过反思和内省的方式进行讲解。尽管这些概念对所有数学家而言都是熟知的，但它们的呈现方式对大多数大学生来说被认为过于新颖，同时从今天的角度来看又显得颇为过时。课程强调了基础概念的相似性，并关注于具体空间而非它们之间的函数。课程还将引入范畴论的视角，但重点是经典视角下基础的紧密性。 课程将详细审视一些早期的基本拓扑空间示例，包括积空间、商空间和子空间等的定义与拓扑性质分析。同时，还讨论连续函数、闭集、开集、豪斯多夫空间、T1空间、极限点、基、基底和次基等主题。需注意的是，课程暂未涵盖米氏空间及米氏拓扑、连接性和紧致性。 本课程面向初学者，尤其适合对拓扑学有兴趣但在数学方面已有一定基础的学员。
If we have a set of points $X$, how can we make a precise notion of closeness and locality? We can define a notion of distance between individual points and have those notions follow as consequences. However, we can be more subtle and define whats known as a /emph{topology} on this set making $X$ /emph{topological space}, which makes precise those notions of closeness, locality, and therefore the notion of continuity (the preserving of closeness) in $X$ directly. Subsequent notions which can also be represented in this setting are that of connectedness (and therefore disconnectedness), compactness and limits.Look at the beginnings of topology and topological spaces. We cover much of Munkres Chapter 2 and its exercises but with reflection and introspection. The ideas are known by all mathematicians and yet the presentation is considered too new for most university students but at the same time looking back on it now is quite strikingly out of date. The basics are still the same but they appear different, the focus is on the concrete spaces and less on the functions between them. Some perspective is added with category theory in mind but much of it is looking closely at the foundations with a classical perspective.Lots of the earlier basic examples of topological spaces are examined in detail.Product spaces, quotient spaces, subspaces are all defined and examined topologically.Continuous functions, closed sets, open sets, Hausdorf space, T1 space, limit point, basis, base, sub base,Metric spaces and metric topology is currently omitted.Connectedness and compactness is omitted.This is for beginners in topology but not necessarily beginners in mathematics especially if you have not used you mind much before.
