Topology by James Munkres | Prerequisites, Level & TOC

[micromass]

• Author: James Munkres
• Title: Topology
• Amazon link https://www.amazon.com/dp/0131816292/?tag=pfamazon01-20
• Prerequisities: Being acquainted with proofs and rigorous mathematics. An encounter with rigorous calculus or analysis is a plus.
• Level: Undergrad

Table of Contents:

[LIST]
[*] Preface
[*] A Note to the  Reader
[*] General Topology
[LIST]
[*] Set Theory and Logic
[LIST]
[*] Fundamental Concepts
[*] Functions
[*] Relation
[*] The Integers and the Real Numbers
[*] Cartesian Products
[*] Finite sets
[*] Countable and Uncountable Sets
[*] The Principle of Recursive Definition
[*] Infinite Sets and the Axiom of Choic
[*] Well-ordered Sets
[*] The Maximum Principle
[*] Supplementary Exercises: Well-Ordering
[/LIST]
[*] Topological Spaces and Continuous Functions
[LIST]
[*] Topological Spaces
[*] Basis for a Topology 
[*] The Order Topology
[*] The Product Topology on [itex]X\times Y[/itex]
[*] Th  Subspace Topology
[*] Closed Sets and Limit Points 
[*] Continuous Functions
[*] The Product Topology
[*] The Metric Topology
[*] The Metric Topology (continued)
[*] The Quotient Topology
[*] Supplementary Exercises: Topological Groups
[/LIST]
[*] Connectedness and Compactness
[LIST]
[*] Connected Spaces
[*] Connected Subspaces of the Real Line
[*] Components and Local Connectedness
[*] Compact Spaces
[*] Compact Subspaces of the Real Line
[*] Limit point compactness
[*] Local Compactness
[*] Supplementary Exercises: Nets
[/LIST]
[*] Countability and Separation Axioms
[LIST]
[*] The Countability Axioms
[*] The Separation Axioms
[*] Normal Spaces
[*] The Urysohn Lemma
[*] The Urysohn Metrization Theorem
[*] The Tietze Extension Theorem
[*] Imbeddings of Manifolds
[*] Supplementary Exercises: Review of  the Basics
[/LIST]
[*] The Tychonoff Theorem
[LIST]
[*] The Tychonoff Theorem
[*] The stone-cech Compactification
[/LIST]
[*] Metrization Theorems and Paracompactness
[LIST]
[*] Local Finiteness
[*] The Nagata-Smirnov Metrization Theorem
[*] Paracompactness
[*] The Smirnov Metrization Theorem
[/LIST]
[*] Complete Metric Spaces and Function Spaces
[LIST]
[*] Complete Metric Spaces
[*] A Space-Filling Curve 
[*] Compactness in Metric Spaces
[*] Pointwise and Compact Convergence
[*] Ascoli's Theorem
[/LIST]
[*] Baire Spaces and Dimension Theory
[LIST] 
[*] Baire Spaces
[*] A Nowhere-Differentiable Function
[*] Introduction to Dimension Theory
[*] Supplementary Exercises: Locally Euclidean Spaces
[/LIST]
[/LIST]
[*] Algebraic Topology
[LIST]
[*] The Fundamental Group
[LIST]
[*] Homotopy of Paths
[*] The Fundamental Group
[*] Covering Spaces
[*] The Fundamental Group of the Circle
[*] Retractions and Fixed Points
[*] The Fundamental Theorem of Algebra
[*] The Borsuk-Ulam Theorem
[*] Deformation Retracts and Homotopy Type
[*] The Fundamental Group of [itex]S^n[/itex
[*] Fundamental Groups of Some Surfaces
[/LIST]
[*]  Separation Theorems in the Plane
[LIST]
[*] The Jordan Separation Theorem
[*] Invariance of Domain
[*] The Jordan Curve Theorem
[*] Embedding Graphs in the Plane 
[*] The Winding Number of a Simple Closed Curve
[*] The Cauchy Integral Formula
[/LIST]
[*] The Seifert-van Kampen Theorem
[LIST]
[*] Direct Sums of Abelian Groups 
[*] Free Products of Groups
[*] Free Groups
[*] The Seifert-van Kampen Theorem
[*] The Fundamental Group of a Wedge of Circles
[*] Adjoining a Two-cell 
[*] The Fundamental Groups of the Torus and the Dunce Cap
[/LIST]
[*] Classification of surfaces
[LIST]
[*] Fundamental Groups of Surfaces
[*] Homology of Surfaces
[*] Cutting and Pasting
[*] The Classification Theorem
[*] Constructing Compact Surfaces
[/LIST]
[*] Classification of Covering Spacs 
[LIST]
[*] Equivalence of Covering Spaces
[*] The Universal Covering Space
[*] Covering Transformations
[*] Existence of Covering Spaces
[*] Supplementary Exercises: Topological Properties and [itex]\pi_1[/itex]
[/LIST]
[*] Applications to Group Theory
[LIST]
[*] Covering Spaces of a Graph
[*] The Fundamental Group of a Graph
[*] Subgroups of Free Groups
[/LIST]
[/LIST]
[*] Bibliography
[*] Index
[/LIST]

 

[jmjlt88]

This textbook is fantastic! It is well-written and very expository. The first four chapters, coupled with a few chapters from Part II (in particular chapters 9 and 11), provide you with an extremely solid foundation in general topology and a taste of algebraic topology.
It is hard to find a weakness, but without a recent course in Analysis (I took Real Analysis 5 years ago), I would say that the two sections on the Metric Topology can be a bit tedious.To remedy this issue, I supplemented these sections with Bert Mendelson's coverage of metric spaces in his book "Introduction to Topology," which is also absolutely fantastic.