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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:

  1. The union of any collection of sets in T is also in T.
  2. The intersection of any pair of sets in T is also in T.
  3. 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

In particular we have:

Topology, the field of mathematics

Topological spaces show up naturally in 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

See also the article on metrization theorems.

Some useful notions from algebraic topology

Sketchy outline of the deeper theory


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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Last edited December 19, 2001 5:56 am by 160.94.28.xxx (diff)