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.

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 2.1 /Basic Concepts Set Theory/

Basic Notions in Topology

• Chapter 2.2.1 /Metric Spaces/
• Chapter 2.2.2 /Topological Spaces/
• Chapter 2.2.3 /Bases/
• Chapter 2.2.4 /Points in Sets/
• Chapter 2.2.5 /Order/
• Chapter 2.2.6 /Sequences/
• Chapter 2.2.7 /Subspaces/
• Chapter 2.2.8 /Continuity and Homeomorphisms/

Topological Properties

• Chapter 2.3.1 Separation Axioms
• Chapter 2.3.2 Connectedness
• Chapter 2.3.3 Path Connectedness
• Chapter 2.3.4 Local Connectedness
• Chapter 2.3.5 Compactness
• Chapter 2.3.6 /Countability/
• Chapter 2.3.7 /Completeness/ - not a topological property

Constructions

• Chapter 2.4.1 /Closure/
• Chapter 2.4.2 Product Spaces
• Chapter 2.4.3 Quotient spaces

Algebraic Topology

Homotopy

• Chapter 3.1 /Free group and presentation of a group/
• Chapter 3.2 /The fundamental group/

Polytopes

• /Barycentric Coordinates/
• /Geometric Complexes/
• /Barycentric Subdivision/
• /Simplical Mappings/
• /Imbedding Theorem/

Homology

• /Relative Homology/
• /Exact Sequences/
• /Mayer-Victoris Sequence/
• /Eilenburg-Steenrod/
• /Excision Theorem/
• /Relative Homotopy/
• /Cohomology/
• /Cohomology Product/
• /Cap-Product/
• /Relative Cohomology/
• /Induced Homeomorphism/
• /Singular Homology/
• /Victoris Homology/
• /Čech Homology/

Differential Topology

• Chapter 4.1 /Manifolds/
• Chapter 4.2 /Tangent Spaces/
• Chapter 4.3 /Vector Bundles/

Help

Question & Answer

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

1. What is the difference between topology, algebra and analysis?

• Topology is a generalization of analysis and geometry. It comes in two main flavors: point-set topology and algebraic topology. The former generalizes concepts from analysis dealing with space such as continuity of functions, connectedness of a space, open and closed sets, (etc.). The latter attributes algebraic structures (groups, rings etc.) to families of topological spaces to distinguish topological differences in those families. A naive description of topology is that it identifies those qualities of a space that do not change under twisting and stretching of that space. As such, it is popularly refered to as "rubber sheet geometry." In reality topology does far more than this, in fact providing a rigorous foundation under all branches of mathematics dealing with "spaces." Differential topology is a sort of hybrid between algebraic topology and geometry, where distance is still irrelevant, but "smoothness" of functions and surfaces is detectable (and required).

• Algebra deals with binary operations on sets, where two elements are combined to make one (such as sums and products). Major areas of interest in algebra are group theory, ring theory and field theory. Each of these focuses on the cases where the operation obeys their respective qualities.

• Analysis (or specifically real analysis) on the other hand deals with the real numbers $ℝ$ and the standard topology and algebraic structure of $ℝ$.

2. How are the concepts of base and open cover related? It seems that every base is an open cover, but not every open cover is a base. But, why are both concepts needed?

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

Marvin Greenberg and John Harper; Algebraic Topology (1981)

Allen Hatcher, Algebraic Topology (2002) [1]

Hu, Sze-tsen, Cohomology Theory (1968)

Hu, Sze-tsen, Homology Theory (1966)

Hu, Sze-tsen, Homotopy Theory (1959)

Albert T. Lundell and Stephen Weingram, The Topology of CW Complexes (1969)

Joerg Mayer, Algebraic Topology (1972)

James Munkres, Elements of Algebraic Topology (1984)

Joseph J. Rotman, An Introduction to Algebraic Topology (1988)

Edwin Spanier, Algebraic Topology (1966)

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