# 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.

## Contents |

## Before You BeginEdit

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 PreliminariesEdit

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

## General TopologyEdit

### Introduction to TopologyEdit

- 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 SpacesEdit

- 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 SpacesEdit

- 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 TopologyEdit

### HomotopyEdit

- 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

### PolytopesEdit

- 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

### HomologyEdit

- 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

### CohomologyEdit

- 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 MethodsEdit

- Chapter 8.1 Poincaré Duality
- Chapter 8.2 Spectral Sequences

## Differential TopologyEdit

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

## AppendicesEdit

- A Further Reading
- B Index

## HelpEdit

### Question & AnswerEdit

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 and the standard topology and algebraic structure of . 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.

### Further ReadingEdit

#### General TopologyEdit

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 TopologyEdit

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)