General Topology

Author:              Zhiming Ou

                         Mattermatics Learning Center

Publisher:       3265 Public Way

ISBN:978-1-0577470-2-2

Summary

Topology is the branch of mathematics that studies the properties of geometric objects that do not rely on the exact shape. It deals with topological Spaces, which is a set of points along with a structure called topology. Topology is constructed on 3 axioms of open sets. A logic system with such a few contents should not be a subject of study, however, just like a blank sheet of paper, it can be filled with any content if you like. Topology can enclose any mathematical structure.

The 4 analysis subjects in mathematics: Mathematical analysis, Real analysis, Complex analysis, and Functional analysis, are fully developed in finite-dimensional spaces. The concept of point limits and body limits, are generalized to net limits in topology; combined with differential geometry, Space analysis becomes possible. If combined with group theory, algebraic topology is born.

This book summarizes the results from geometric topology and point-set topology. Various spaces are studied. The theory can be used for machine proofs of geometric theorems, as well as the quantum field theory.

 

Content

Introduction: what is topology?          4

Chapter 1 Surface Topology              8

  • 1 Homeomorphism (同胚) and Homotopism（同调）
  • 2 Manifolds 9

       Charts and Atlas     9

       Cartesian Products       11

  • 3 Topological Properties 12

Order of Connectivity         12

Orientability        13

Genus        16

Euler Characteristic        19

Betti Numbers       19

  • 4 The Klein Bottle 21

 

Chapter 2 Point-set Topology         28

  • 1 Definitions 28
  • 2 Closed subsets of a topological space 31
  • 3 Set closures, interiors and boundaries 33

Dense subsets of a topological space       41

Countable dense subsets        42

Regular open sets and regular closed sets     42

  • 4 Bases of topological spaces 43

Neighborhoods of points     43

A base for a topology       44

The subbase of a topology        47

  • 5 Spaces with countable bases 49
  • 6 Hereditary topological properties 52

 

Chapter 3 Construction of Topological Spaces        53

  • 1 Four Ways to construct a topological space 53

Relative Topology / Subspace Topology

Free Union of Topological Spaces

Product Topology       55

Quotient Topology        62

  • 2 Examples of Topological Spaces 67

Norms on Vector Spaces      67

The Metric Space         69

The Ordinal Space          75

  • 3 Function Spaces 77

The embedding theorems

Metric in C(S)      81

Some Theorems from Functional Analysis        83

 

Chapter 4 Limit Points in a Topological Space        86

  • 1 limit of sequence 86
  • 2 Limit points of nets 89
  • 3 Convergence of functions 100
  • 4 Limit points of filters 104

 

Chapter 5 Separation Axioms         118

  • 1 Separation with Open Sets 118

The T0 separation axiom       

The T1 separation axiom          119

T2 or Hausdorff Spaces        120

 or Completely Hausdorff Spaces          122

T3-spaces and “regular spaces”          123

T4-spaces and normal spaces                124

  • 2 Separations with Continuous Functions 126

Normal Spaces    126

Completely Regular Spaces       129

zero-sets in normal spaces          131

Perfectly normal topological spaces       132

 

Chapter 6 Compact Spaces and Relatives         137

  • 1 Concepts and Properties 137
  • 2 Countably compact spaces 145
  • 3 Lindelof Spaces 146
  • 4 Sequentially compact and pseudocompact spaces 148
  • 5 Locally Compact Spaces 149
  • 6 Paracompact topological spaces 151

 

Chapter 7 Connected Spaces        154

  • 1 Connected set and space 154
  • 2 Locally connected: Spaces with an open base of connected sets 161
  • 3 Pathwise connected spaces 163
  • 4 Totally disconnected spaces 167

 

Chapter 8 Complete Spaces        171

  • 1 Metrics on sets 171
  • 2 Complete Spaces 176

Baire Spaces   178

 

Chapter 9 Uniform Spaces      180