13.08.2022, 18:47
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Topological Spaces Starter
MP4 | Video: h264, 1280x720 | Audio: AAC, 44.1 KHz
Language: 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 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is considered too new for most university students but at the same time looking back on it now is quite strikingly out of date.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.
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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