Topology

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 generalizes many distance-related concepts, such as continuity, compactness, and convergence.

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.

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.

Motivation and Preliminaries

Chapter 1.1 Introduction
Chapter 1.2 History
Chapter 1.3 Basic Concepts Set Theory

General Topology

Introduction to 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 Sequences
Chapter 2.2.6 Subspaces
Chapter 2.2.7 Order
Chapter 2.2.8 Order Topology
Chapter 2.2.9 Product Spaces
Chapter 2.2.10 Quotient Spaces
Chapter 2.2.11 Continuity and Homeomorphisms

Properties of Topological Spaces

Chapter 2.3.1 Separation Axioms
Chapter 2.3.2 Connectedness
Chapter 2.3.3 Path Connectedness
Chapter 2.3.4 Compactness
Chapter 2.3.5 Comb Space
Chapter 2.3.6 Local Connectedness
Chapter 2.3.7 Linear Continuum
Chapter 2.3.8 Countability
Chapter 2.3.9 Cantor Space
Chapter 2.3.10 Completeness - not a topological property
Chapter 2.3.11 Completion
Chapter 2.3.12 Perfect map - optional section which is challenging

Vector Spaces

Chapter 3.1 Vector Spaces
Chapter 3.2 Morphisms
Chapter 3.3 Convexity
Chapter 3.4 Hahn-Banach Theorem
Chapter 3.5 Topological Vector Spaces
Chapter 3.6 Euclidean Spaces
Chapter 3.7 Normed Vector Spaces
Chapter 3.8 Banach Spaces
Chapter 3.9 Hilbert Spaces

Algebraic Topology

Homotopy

Chapter 4.1 Free group and presentation of a group
Chapter 4.2 Deformation Retract
Chapter 4.3 Homotopy
Chapter 4.4 The fundamental group
Chapter 4.5 Induced homomorphism

Polytopes

Chapter 5.1 Simplicial complexes
Chapter 5.2 Barycentric Coordinates
Chapter 5.3 Geometric Complexes
Chapter 5.4 Barycentric Subdivision
Chapter 5.5 Simplical Mappings
Chapter 5.6 Imbedding Theorem

Homology

Chapter 6.1 Exact Sequences
Chapter 6.2 Homology Groups
Chapter 6.3 Singular Homology
Chapter 6.4 Relative Homology
Chapter 6.5 Mayer-Vietoris Sequence
Chapter 6.6 Excision Theorem
Chapter 6.7 Eilenberg–Steenrod axioms
Chapter 6.8 Relative Homotopy
Chapter 6.9 Vietoris Homology

Cohomology

Chapter 7.1 Cohomology
Chapter 7.2 Cohomology Product
Chapter 7.3 Cap-Product
Chapter 7.4 Relative Cohomology
Chapter 7.5 Induced Homeomorphism
Chapter 7.6 Čech Cohomology

Advanced Methods

Chapter 8.1 Poincaré Duality
Chapter 8.2 Spectral Sequences

Differential Topology

Chapter 9.1 Manifolds
Chapter 9.1.1 Categories of Manifolds
Chapter 9.2 Tangent Spaces
Chapter 9.3 Vector Bundles

Appendices

A Further Reading
B Index

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 many flavors: point-set topology, manifold topology and algebraic topology, to name a few. All topology generalizes concepts from analysis dealing with space such as continuity of functions, connectedness of a space, open and closed sets, (etc.). Algebraic topology attributes algebraic structures (groups, rings etc.) to families of topological spaces to distinguish topological differences in those families. Manifold topology works with spaces that are locally the same as Euclidean space, i.e., surfaces. Often manifolds are equipped with extra structure, such as smooth, PL, symplectic, etc. 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 referred 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."

Algebra deals with the structure of sets under various operations with particular properties. Commonly studied algebraic objects include Groups, Rings and Field. One of the major results from Algebra include Galois Theory, which eventually shows that there is no general solution to quintic polynomial equations by radicals. Also important results from Algebra are the Fundamental Theorem of Algebra (which says that, in the Field of Complex numbers, every non constant polynomial has at least one root), Group Classification, and much more.

Analysis (or specifically real analysis) on the other hand deals with the real numbers $\mathbb {R}$ and the standard topology and algebraic structure of $\mathbb {R}$ . Analysis provides rigorous proofs for the definitions of derivatives and integrals, as well as treatments of sequences and limits. One can, in some sense, view it as a rigorous treatment of the Calculus.

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?

The terms base and open cover are not evidently related. Every base is an open cover which is probably the main relation. Take a second countable topological space for instance (second countable means that the space has a countable base for its topology). Such a space satisfies the property that every open cover has a countable sub-cover. To prove this we use the countability of the base. Basically, for any open cover, we choose for each element of the space, an element of the open cover containing it and hence a basis element contained in that element of open cover. Therefore, for any open cover, we can generate a open cover of basis elements that is an 'open refinement' (see Wikipedia for definition). From here we can get properties of open covers from properties of the base. If the base is countable, we can generate a countable open cover from the original cover.

The reason we have both definitions is because these two things have different properties. The most useful fact about a base is that it determines the topology. A basis must have "arbitrarily small" sets, that is, any open set contains a basis element. On the other hand, an open cover does not determine the topology at all. It can be used to build things such as partitions of unity, and often draws on the compactness property. Topology Expert (talk) 04:17, 8 June 2008 (UTC)

3. What is a homology?

While homotopy is what one would want to study at first, it turns out that most questions in homotopy theory are rather hard. If we "replace" the spheres of homotopy theory with simplices we can extract similar information about "holes" in the space (often what one is interested in), we get a much more computable sequence of groups. The Hurewicz theorem even gives us that in some cases the homotopy groups can be calculated via the homology groups.

Topology

Before You Begin

Motivation and Preliminaries

General Topology

Introduction to Topology

Properties of Topological Spaces

Vector Spaces

Algebraic Topology

Homotopy

Polytopes

Homology

Cohomology

Advanced Methods

Differential Topology

Appendices

Help

Question & Answer

Further Reading

General Topology

Algebraic Topology