Branches of Mathematics
Topology
Also Known As Rubber-sheet Geometry
Citation Formats
General Reference
APA Style
BibTeX
Sometimes called rubber-sheet geometry: the study of properties of a space that survive continuous deformation, stretching and bending but not tearing or gluing. A coffee cup and a doughnut are topologically the same shape, each with exactly one hole. The field grew out of Leonhard Euler's 1736 solution to the Seven Bridges of Konigsberg problem and matured through the twentieth century into algebraic and differential topology, culminating in results such as the 2003 proof of the Poincare conjecture.
Facts
Central QuestionWhich properties of a space survive continuous deformation, and how can spaces be classified by those properties alone, independent of any notion of distance or angle? 1 Key DebateWhether topology is fundamentally about the intuitive picture of rubber-sheet shape, or whether, as the categorical reframing of the subject holds, it is really about structure-preserving maps between spaces, a description under which the intuitive picture of stretching and bending is only a special, if illuminating, case. 1 Cross-Tradition Connections
Sources
1. MacTutor History of Mathematics Archive
University of St Andrews, School of Mathematics and Statisticshttps://mathshistory.st-andrews.ac.uk/HistTopics/Topology_in_mathematics/Quote, https://mathshistory.st-andrews.ac.uk/HistTopics/Topology_in_mathematics/
Topological ideas are present in almost all areas of today's mathematics. The subject of topology itself consists of several different branches, such as point set topology, algebraic topology and differential topology, which have relatively little in common.
1. MacTutor History of Mathematics Archive
University of St Andrews, School of Mathematics and StatisticsAssociated With: Henri Poincare, https://mathshistory.st-andrews.ac.uk/Biographies/Poincare/Quote, Associated With: Henri Poincare, https://mathshistory.st-andrews.ac.uk/Biographies/Poincare/
Henri Poincare can be said to have been the originator of algebraic topology and of the theory of analytic functions of several complex variables.
Topology (Britannica)
Encyclopaedia Britannica, Inc.opening definitionQuote, opening definition
Topology, branch of mathematics, sometimes referred to as rubber sheet geometry, in which two objects are considered equivalent if they can be continuously deformed into one another.
Wikipedia: Topology
Wikimedia FoundationLead section, opening paragraphQuote, Lead section, opening paragraph
the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations
Brouwer Fixed-Point Theorem (Wikipedia)
Wikimedia FoundationIncludes: Brouwer Fixed-Point Theorem, opening sentenceQuote, Includes: Brouwer Fixed-Point Theorem, opening sentence
It states that for any continuous function f mapping a nonempty compact convex set to itself, there is a point x0 such that f(x0) = x0.
Metric Space (Wikipedia)
Wikimedia FoundationIncludes: Metric Space, lead paragraphQuote, Includes: Metric Space, lead paragraph
a metric space is a set together with a notion of distance between its points
Manifold (Wikipedia)
Wikimedia FoundationIncludes: Manifold, lead paragraphQuote, Includes: Manifold, lead paragraph
a manifold is a topological space that locally resembles Euclidean space near each point
Michael Atiyah (Wikipedia)
WikipediaIncludes: Michael Atiyah, Lead sectionQuote, Includes: Michael Atiyah, Lead section
His contributions include the Atiyah-Singer index theorem and co-founding topological K-theory.
Poincare Conjecture (Wikipedia)
Wikimedia FoundationIncludes: Poincare Conjecture, IntroductionQuote, Includes: Poincare Conjecture, Introduction
In the mathematical field of geometric topology, the Poincare conjecture is a theorem about the characterization of the 3-sphere.
Reader Challenges (0 open reader challenges)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.
View At A Past Year
The atlas records no dated fact of its own for this entry, so there is no other year to choose.