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.

History

It could be said that mathematics in general owes its credibility to ancient Greece's Euclid. What is probably his most famous work, Elements, revolutionized the concepts of geometry and mathematics as a whole through the presentation of a simple logical method. This method is summarized by Leonard Mlodinow:

First, make terms explicit by forming precise definitions and so ensure mutual understanding of all words and symbols. Next, make concepts explicit by stating specific axioms or postulates so that no unstated understandings or assumptions may be used. Finally, derive the logical consequences of the system employing only accepted rules of logic, applied to the axioms and to previously proved theorems .

Throughout its history, many mathematicians have influenced the development of topology. While Johann Benedict Listing is not credited with a memorable discovery in terms of the field of topology, he is still considered one of the founding fathers. This is because he gave topology its name. While he published very little on topology, he is remembered for Vorstudien zur Topologie, which was the first document to use the word topologie (English: topology) to describe the field. He is also often credited with discovering the Möbius strip independently of August Ferdinand Möbius .

The origins of topology date back to the eighteenth century and the Königsberg Bridge Problem, a problem of relative position without regard to distance . While this problem is often regarded as the birth of graph theory, it also inspired Euler's development of the topology of networks . Königsberg, now Kaliningrad, was founded in 1255 and became a prosperous seaport . The city resides on the banks of the Praegel, now Pregolya, River. Citizens could use seven bridges that crossed the Praegal, but the question of whether or not one could pass through the town and use each bridge exactly once would turn out to be the catalyst in the creation of the mathematical field of topology. Swiss mathematician Leonhard Euler would be the one to discover the answer was no. He determined that the graph defined by the location of the bridge was not what is now called a Eulerian graph . This solution entitled The Solution of a Problem Related to the Geometry of Position was submitted to the Academy of Sciences in St. Petersburg in 1735 .

The Königsberg Bridge Problem.

Mobius Strip

Euler is also well known for his research in the combinatorial qualities of polyhedra. He considered the edges ($e$), which he called acies, the faces ($f$), or hedra, and the vertices ($v$), called angulus solidus. Euler realized the importance of these three properties claiming that they "completely determine the solid". His research resulted in the well-known Polyhedral Formula: $v-e+f=2$. However, Euler's formula applies only to convex solids . In 1813 Antoine-Jean L'Huilier recognized this limitation of the formula and provided a generalization for a solid with $g$ holes: $v-e+f=2-2g$. This was the first known result of a topological invariant .

August Ferdinand Möbius was one of the main contributors of the topological theory of manifolds. In 1865, Möbius presented an article in which he decomposed several orientations of surfaces in polygonal nets. His most famous example was a non-orientable surface, which is now called the Möbius strip.

The Russian born mathematician Georg Ferdinand Ludwig Philipp Cantor, the father of set theory, is another mathematician to whom we owe credit for topology. Concepts of set theory and cardinality are fundamental for the study of topology. Today, Cantor is a truly celebrated mathematician, especially considering that set theory and the idea of infinity do not seem to have a truss of mathematical ideas from which they could have been developed. Sadly, these ideas were not welcomed by a nineteenth century world, and Cantor spent many years of his adult life struggling with public criticism. A German mathematician by the name of David Hilbert described Cantor's discoveries in the infinite domain as an "astonishing product of mathematical thought" . In 1877, Cantor showed that the points on a 2-dimensional square had a one-to-one correspondence with the points on different line segments, and this caused others to begin asking questions about the idea of dimension, leading to the development of dimension theory .

In the late 1800s and early 1900s, many mathematicians challenged themselves with more abstract problems. Maurice René Fréchet, a French mathematician, helped these mathematicians considerably in 1906. He explained that if a distance can be defined between two different mathematical entities, then real and complex number concepts can be applied . Fréchet, along with Schoenflies, Hausdorff, and others, would be one of the first to study "general topology" . Fréchet developed the theory of metric spaces, which was based on Cantor's theory of sets .

German mathematician Felix Hausdorff followed in Cantor's footsteps with regard to set theory. In fact, Hausdorff was one of the first to teach set theory. In the summer 1901, he had 3 students . The idea that a topology possesses a lattice of open subsets had been around almost as long as the idea of topology itself, but Hausdorff was the first to emphasize the importance of these sets in defining topological concepts .

French mathematician and physicist Henri Poincaré discovered his talent at an early age. In fact, he took first place in a national mathematics competition while he was still in school. Poincaré was the first to study Fuchsian groups, dealing mainly with their underlying geometry and topology . Poincaré is most famous for The Poincaré Conjecture which states the following:

A compact smooth n-dimensional manifold that is homotopy equivalent to the n-sphere $S^n$ must in fact be homeomorphic to $S^n$. One can think of a compact manifold as a manifold that lives in a finite region of $\mathbb{R}^n$ for some $n$ and that has no boundary .

This conjecture would not be proven until 2003 by Grigory Perelman .

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.

Point - Set Topology

Some Set Theory

• Chapter 2.1 Basic Concepts Set Theory

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 Null Space
• Chapter 3.4 Hyperplanes
• Chapter 3.5 Convexity
• Chapter 3.6 Hahn-Banach Theorem
• Chapter 3.7 Normed Vector Spaces
• Chapter 3.8 Euclidean Spaces
• Chapter 3.9 Hilbert Spaces
• Chapter 3.10 Topological Vector Spaces

Algebraic Topology

Homotopy

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

Polytopes

• Chapter 5.1 Simplicial complexes
• Barycentric Coordinates
• Geometric Complexes
• Barycentric Subdivision
• Simplical Mappings
• Imbedding Theorem

Homology

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

Differential Topology

• Chapter 6.1 Manifolds
• Chapter 6.1.1 Categories of Manifolds
• Chapter 6.2 Tangent Spaces
• Chapter 6.3 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 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?

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)

Index

Index