Topology - eMyTextBooks

Topology

Introduction

General Topology is based solely on set theory and concerns itself with structures of sets. It is at its core a generalization of the concept of distance, though this will not be immediately apparent for the novice student. Topology generalises many distance related concepts, such as continuity, compactness and convergence.

For an overview of the subject of topology, please see the Wikipedia entry (http://en.wikipedia.org/wiki/Topology).

Before You Begin

In order to make things easier for you as a reader, as well as for the writers, you will be expected to be familiar with a few topics before beginning. (I hope to have some links to other Wikibooks here soon.)

• Real analysis
  • Continuous Functions
  • Sequences & Series, Convergence & Divergence
• Set Theory
  • Set Operations: Union, Intersection, Complement, De Morgan's laws, etc.
  • Order Relations: Ordered Sets, Equivalence relations, Lattices.
  • Functions: Definition and Properties of Functions
  • Cardinality: Finite, Countable, and Uncountable sets
  • Zorn's Lemma and the Axiom of Choice
• Mathematical Logic & Proofs
  • Mathematics is all about proofs. One of the goals of this book is to improve your skills in doing proofs, but you will not learn any of the basics here.

Point - Set Topology

Some Set Theory

• Chapter 1 Basic Concepts Set Theory

Basic Notions in Topology

• Chapter 2.1 Topological Spaces
• Chapter 2.2 Bases
• Chapter 2.3 Metric Spaces
• Chapter 2.4 Order
• Chapter 2.5 Sequences
• Chapter 2.6 Subspaces
• Chapter 2.7 Points in Sets
• Chapter 2.8 Continuity and Homeomorphisms

Topological Properties

• Chapter 3.1 Separation Axioms
• Chapter 3.2 Connectedness
• Chapter 3.3 Path Connectedness
• Chapter 3.4 Compactness
• Chapter 3.5 Countability

Constructions

• Chapter 4.1 Product Spaces
• Chapter 4.2 Quotient Spaces

Algebraic Topology

Algebraic Topology

• Chapter 5.1 Free group and presentation of a group
• Chapter 5.2 The fundamental group

Differential Topology

Help

Question & Answer

Have a question? Why not ask the very textbook that you are learning from?

!. what is the difference between topology, algebra and analysis

• Topology deals mostly with concepts such as open sets and continuous functions. Algebra deals with binary operations where two elements are combined to make one (such as sums and products). Analysis (or specifically real analysis) on the other hand deals with the real numbers and the standard topology and algebraic structure of .

Further Reading

General Topology

Aleksandrov; Combinatorial Topology (1956)

Baker; Introduction to Topology (1991)

Dixmier; General Topology (1984)

Engelking; General Topology (1977)

Munkres; Topology (2000)

James; Topological and Uniform Spaces (1987)

Jänich; Topology (1984)

Kuratowski; Introduction to Set Theory and Topology (1961)

Kuratowski; Topology (1966)

Roseman; Elementary Topology (1999)

Seebach, Steen; Counterexamples in Topology (1978)

Willard; General Topology (1970)

Algebraic Topology

Greenburg, Harper; Algebraic Topology (1981)

Hatcher; Algebraic Topology (2002)

Hu; Cohomology Theory (1968)

Hu; Homology Theory (1966)

Hu; Homotopy Theory (1959)

Lundell, Weingram; The Topology of CW Complexes (1969)

Mayer; Algebraic Topology (1972)

Munkres; Elements of Algebraic Topology (1984)

Rotman; An Introduction to Algebraic Topology (1988)

Spanier; Algebraic Topology (1966)

External Links