Topology

Topology is that branch of mathematics concerned with the study of topological spaces. (The term topology is also used for a system of open sets used to define topological spaces, but this article will focus on the branch of mathematics. Wiring and computer network topologies are discussed in Network topology.)

Topological spaces show up naturally in mathematical analysis, abstract algebra and geometry. This has made topology one of the great unifying ideas of mathematics. Point-set topology (or general topology) defines and studies some very useful properties of spaces and maps, such as connectedness, compactness and continuity. Algebraic topology is a powerful tool to study topological spaces, and the maps between them. It associates "discrete", more computable invariants to maps and spaces, often in a functorial way. Ideas from algebraic topology have had strong influence on algebra and algebraic geometry.

Please refer to the Topology glossary for the definitions of terms used throughout topology.

Elementary introduction

The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the "way they are connected together". One of the first papers in topology was the demonstration, by Leonhard Euler, that it was impossible to find a route through the town of Königsberg (now Kaliningrad) that would cross each of its seven bridges exactly once. This result did not depend on the lengths of the bridges, nor on their distance from one another, but only on connectivity properties of which bridges are connected to which islands or riverbanks. This problem, the seven bridges of Königsberg, is now a famous problem in introductory mathematics.

Similarly, it is a theorem of algebraic topology (the hairy ball theorem) that "hairy ball theorem|you can't comb the hair on a ball smooth". This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem: There is no nonvanishing continuous tangent vector field on the sphere. As with the bridges of Königsberg, the result doesn't depend on the exact shape of the sphere. It applies to pear shapes and in fact any kind of blob, as long as it has no holes.

In order to deal with these problems that don't rely on the exact shape of the objects, one must be clear about just what properties these problems do rely on. From this need arises the notion of topological equivalence. The impossibility of crossing each bridge just once applies to any arrangement of bridges topologically equivalent to those in Königsburg, and the hairy ball theorem applies to any space topologically equivalent to a sphere.

Formally, two spaces are topologically equivalent if there is a homeomorphism between the two spaces. In that case the spaces are said to be homeomorphic, and in fact the term "topologically equivalent" is mainly used when explaining topology to non-topologists.

Some useful theorems

• Every interval in R is connected.
• Every closed interval in R of finite length is compact. More is true: In Rⁿ, a set is compact iff it is closed and bounded. (See Heine-Borel theorem.)
• A metric space is Hausdorff, also normal and paracompact.
• If X is a complete metric space or a locally compact Hausdorff space, then the interior of every union of countably many nowhere dense sets is empty. (See Baire category theorem.)
• In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map defined on the whole space. (See Tietze extension theorem.)
• The continuous image of a connected space is connected.
• The continuous image of a compact space is compact. (Proof: The image of a finite subcover of the preimage of an open cover of the image is a finite subcover of the open cover of the image.)
• The (arbitrary) product of compact spaces is compact. (See Tychonoff's theorem.)
• A compact subspace of a Hausdorff space is closed.
• Every sequence of points on a compact metric space has a convergent subsequence.
• On a paracompact Hausdorff space every open cover admits a partition of unity subordinate to the cover.

See also the article on metrization theorems.

Some useful notions from algebraic topology

• Homology and cohomology: Betti numbers, Euler characteristic.
• Nice applications: Brouwer Fixed Point Theorem, Borsuk-Ulam Theorem
• Homotopy groups (including the fundamental group.)
• Chern classes, Stiefel Whitney classes, Pontrjagin classes

Sketchy outline of the deeper theory

• (Co)fibre sequences: Puppe sequence, computations
• Homotopy groups of spheres
• Obstruction theory
• K-theory: KO, algebraic K-theory
• Stable homotopy
• Brown representability
• (Co)bordism
• Signatures
• BP and Morava K-theory
• Surgery obstructions
• H-spaces, infinite loop spaces, A_∞ rings
• Homotopy theory of schemes
• Intersection cohomology

Generalizations

Occasionally, one needs to use the tools of topology but a "set of points" is not available. In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are certain structures defined on arbitrary categories which allow the definition of sheaves on those categories, and with that the definition of quite general cohomology theories.

See also

• Topological space
• Network topology
• Link topology
• Topology of the universe

External link

• ODP category