Topological Spaces Starter

[LeeAndro]

MP4 | Video: h264, 1280x720 | Audio: AAC, 44.1 KHzLanguage: English | Size: 7.09 GB | Duration: 15h 51m

Topology introduction revisited

What you'll learn
Define a topological space, topology, open set, closed set, continuous function
Understand and use the notion of a base or basis of neighbourhoods
Various universal constructions including product, quotient and subspace topology
Definition chase, basic proofs, solve simple exercises with unfamiliar terms
Requirements
Growth mindset
Abstract thinking and thoughtfulness
Decent short term and long term memory
Experience in abstract algebra and other high level maths will be helpful but not essential
Experience in analysis helpful but not essential
Experience in calculus helpful but not essential
Experience in Linear algebra helpful but not essential
Description
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 bnings 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is considered too new for most university students but at the same looking back on it now is quite strikingly out of date.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 bners in topology but not necessarily bners in mathematics especially if you have not used you mind much before.

Overview

Section 1: Introduction

Lecture 1 Introduction

Section 2: Definitions

Lecture 2 Topology

Lecture 3 Bases

Lecture 4 Subbasis

Section 3: Exercise set 1

Lecture 5 Getting started

Lecture 6 Review of first section plus some exercises analysed

Lecture 7 Exercise 5 Yasiru

Lecture 8 Exercise 5 continued

Lecture 9 Exercise 6 Yee

Lecture 10 Exercise 7

Section 4: Examples of topologies

Lecture 11 Order topology

Lecture 12 Examples

Lecture 13 Product

Section 5: Exercise set 2

Lecture 14 Exercise 1

Section 6: Closure via exercises

Lecture 15 Closure exercise sample

Section 7: Homeomorphisms

Lecture 16 Continuous functions

Lecture 17 Homeomorphisms

Section 8: Exercises

Lecture 18 Continuous image of limit point

Lecture 19 Continuous at 1 point only

Section 9: Group theory Intermission

Lecture 20 Orbit stabiliser theorem

Section 10: Quotients

Lecture 21 Universal properties

Lecture 22 Quotient spaces

Section 11: Hausdorff spaces

Lecture 23 T1, limit points

Lecture 24 Hausdorff basics

Section 12: Extras

Lecture 25 Topological groups preview

Students who are trying to grasp abstractions in maths at a high level,Students who want to fill in gaps from their knowledge,Students who want to be mathematicians,Smart students

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Https://anonymz.com/https://www.udemy.com/course/topology

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