Topology

Topology, in mathematics, is both a structure used to capture the notions of continuity, connectedness and convergence, and the name of the branch of mathematics which studies these.

Topology, the structure

Formally, a topology for a set X is defined as a set T of subsets of X (i.e., T is a subset of the power set of X) satisfying:

The union of any collection of sets in T is also in T.
The intersection of any pair of sets in T is also in T.
X and the empty set are in T.

The sets in T are referred to as open sets, and their complements in X are called closed sets. Roughly speaking, open sets are thought of as neighborhoods of points; two points are "close together" if there are many open sets that contain both of them.

There are many other equivalent ways to define a topology. Instead of defining open sets, it is possible to define first the closed sets, with the properties that the intersection of closed sets is closed, the union of a finite number of closed sets is closed, and X and the empty set are closed. Open sets are then defined as the complements of closed sets. Another method is to define the topology by means of the closure operator. The closure operator is a function from the power set of X to itself which satisfies the following axioms (called the Kuratowski closure axioms): the closure operator is idempotent, every set is a subset of its closure, the closure of the empty set is empty, and the closure of the union of two sets is the union of their closures. Closed sets are then the fixed points of this operator.

A set together with a topology for the set is called a topological space. A function between topological spaces is said to be continuous if the inverse image of every open set is open.

A great many terms are used in topology. Some of these terms have been collected together in the Topology Glossary, and the rest of this article assumes that the reader is familiar with them.

Examples of topological spaces

Any set with the discrete topology (i.e., every set is open, which has the effect that no two points are "close" to each other).
Any set with the trivial topology (i.e., only the empty set and the whole set are open, which has the effect of "lumping all points together").
Any infinite set with the cofinite topology (i.e., the open sets are the empty set and the sets whose complement is finite). This is the smallest T₁ topology on the set.
The real numbers R: the open sets are unions of (possibly infinitely many) open intervals. This is in many ways the most basic topological space and the one that guides most of our human intuition.
The complex numbers C: the open sets are unions of open discs.
A subset of a topological space. The open sets are the intersections of the open sets of the larger space with the subset. This is also called a subspace.
Any metric space turns into a topological space if we define a set to be open if it is a (possibly infinite) union of open balls.

In particular we have:

Intervals in R.
Products of topological spaces. For finite products, the open sets are the sets that are unions of products of open sets.
In particular, Rⁿ (see Euclidean space) and closed cubes or equivalently closed balls, often called cells.
Manifolds. In particular, surfaces.
A simplex. Convex objects that are very useful in computational geometry. In 0, 1, 2 and 3 dimensional space the simplexes are the point, line segment, triangle and tetrahedron, respectively.
Simplicial complexes. A Simplicial Complex is made up of many Simplices. Many geometric objects can be modeled by Simplicial Complexes - see also polytope.
Quotients. If f: X -> Y is a map and X is a topological space, then Y gets a topology where a set is open if and only if its inverse image is open.
CW complexes. Glued together cells that inherit essentially the quotient topology.
Metric spaces. Spaces where the distance between points is defined. This includes useful infinite dimensional spaces like Banach spaces and Hilbert Space studied in functional analysis.
The Zariski topology. A purely algebraically defined topology on the spectrum of a ring or a variety. On Rⁿ or Cⁿ the closed sets of the Zariski topology are just the zerosets of polynomial equations.
The weak topology. A useful topology for operators studied in functional analysis.

Topology, the field of mathematics

Topological spaces show up naturally in mathematical analysis, 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. 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 (see category theory). Ideas from algebraic topology have had strong influence on algebra and algebraic geometry.

Some useful theorems

Every closed interval is compact. More is true: In Rⁿ , a set is compact iff it is closed and bounded. (Theorem of Heine-Borel).
Every closed interval is connected.
A metric space is Hausdorff, also normal and paracompact.
If X is a complete metric space or locally compact Hausdorff, then the complement of every countable union of nowhere dense sets is dense. (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. (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 the finite subcover of the preimage of the open cover of the image is a finite subcover of the open cover of the image.)
The (arbitrary) product of compact spaces is compact. (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 covering 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_infty rings
homotopy theory of schemes
Intersection Cohomology

Generalizations

Occasionly, 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.

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Wiring and computer network topologies are discussed in Network topology