From Wikipedia, the free encyclopedia


TrefoilKnot 02.svg

Topology (Greek topos, "place," and logos, "study") is a branch of mathematics that is an extension of geometry. Topology begins with a consideration of the nature of space, investigating both its fine structure and its global structure. Topology builds on set theory, considering both sets of points and families of sets.

The word topology is used both for the area of study and for a family of sets with certain properties described below that are used to define a topological space. Of particular importance in the study of topology are functions or maps that are homeomorphisms. Informally, these functions can be thought of as those that stretch space without tearing it apart or sticking distinct parts together.

When the discipline was first properly founded, toward the end of the 19th century, it was called geometria situs (Latin geometry of place) and analysis situs (Latin analysis of place). From around 1925 to 1975 it was an important growth area within mathematics.

Topology is a large branch of mathematics that includes many subfields. The most basic division within topology is point-set topology, which investigates such concepts as compactness, connectedness, and countability; algebraic topology, which investigates such concepts as homotopy and homology; and geometric topology, which studies manifolds and their embeddings, including knot theory.

Selected article

A 3-D depiction of a trefoil knot

Knot theory is the mathematical branch of topology that studies mathematical knots, which are defined as embeddings of a circle in 3-dimensional Euclidean space, R3. This is basically equivalent to a conventional knotted string with the ends joined together to prevent it from becoming undone. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself. These transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

Knots can be described in various ways. Given a method of description, however, there may be more than one description that represents the same knot. Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot. One way of distinguishing knots is by using a knot invariant, a "quantity" which remains the same even with different descriptions of a knot.

...Archive Image credit: User:Jtico


The Mathematics WikiProject is the center for mathematics-related editing on Wikipedia. Join the discussion on the project's talk page.


Project pages



Related projects

Selected picture

Mug and torus
Credit: User:LucasVB

It is often suggested that a topologist cannot tell the difference between a coffee cup and a doughnut. This is because these objects when thought of as topological spaces are homeomorphic. The above picture depicts a continuous deformation of a coffee cup into a doughnut such that at each stage the object is homeomorphic to the original.

...Archive

Did you know?

DYK Question Mark Right


Topics in Topology

Main articles Key concepts Algebraic topology Geometric topology

Related portals


Topology on Wikinews     Topology on Wikiquote     Topology on Wikibooks     Topology on Wikisource     Topology on Wiktionary     Topology on Wikimedia Commons
News Quotations Manuals & Texts Texts Definitions Images & Media

Purge server cache

Retrieved from "https://en.wikipedia.org/w/index.php?title=Portal:Topology&oldid=643806234"
This content was retrieved from Wikipedia : http://en.wikipedia.org/wiki/Portal:Topology
This page is based on the copyrighted Wikipedia article "Portal:Topology"; it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License (CC-BY-SA). You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA