Portal:Topology

From Wikipedia, the free encyclopedia

Culture · Geography · Health · History · Mathematics · Natural sciences · Philosophy · Religion · Society · Technology

edit  

Topology

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.

Read more Show new selections edit  

Selected article

The homotopy groups of spheres describe the different ways spheres of various dimensions can be wrapped around each other. They are studied as part of algebraic topology. The topic can be hard to understand because the most interesting and surprising results involve spheres in higher dimensions. These are defined as follows: an n-dimensional sphere, n-sphere, consists of all the points in a space of n+1 dimensions that are a fixed distance from a center point. This definition is a generalization of the familiar circle (1-sphere) and sphere (2-sphere).

A homotopy from a circle around a sphere down to a single point.

The goal of algebraic topology is to categorize or classify topological spaces. Homotopy groups were invented in the late 19th century as a tool for such classification, in effect using the set of mappings from an n-sphere in to a space as a way to probe the structure of that space. An obvious question was how this new tool would work on n-spheres themselves. No general solution to this question has been found to date, but many homotopy groups of spheres have been computed and the results are surprisingly rich and complicated. The study of the homotopy groups of spheres has led to the development of many powerful tools used in algebraic topology.

...Archive Image credit: Richard Morris Read more...
edit  

WikiProjects

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

Project pages

Subprojects

Related projects

Computer science | Cryptography | Game theory | Numbers | Physics | Science edit  

Selected picture

Mug and torus Credit: User:Kieff

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 Read more...
edit  

Did you know?

edit  

Categories

Algebraic topology • Continuous mappings • Differential topology • Fiber bundles • Fixed points • General topology • Geometric group theory • Geometric topology • Homeomorphisms • Limit sets • Molecular topology • Network topology • Orientation • Sheaf theory • Topological algebra • Topological graph theory • Topological spaces • Topologists • Uniform spaces • Topology stubs
edit  

Topics in Topology

Main articles Key concepts Algebraic topology Geometric topology
  • Topological spaces
  • General (point set) topology
  • Differential topology
  • Combinatorial topology
  • Metric topology
  • Set-theoretic topology
  • Topological properties
  • Separation axioms
  • List of examples in general topology
  • Glossary of general topology
  • List of general topology topics
  • Open set, closed set
  • Continuity
  • Compact space
  • Uniform space
  • Metric space
  • Hausdorff space
  • Simplicial complexes
  • CW complexes
  • Exact sequence
  • Homological algebra
  • K-theory
  • Geometric topology
  • Low-dimensional topology
  • Knot theory
  • Braid group
  • Genus
  • Topological manifold
    • 3-manifold
    • 4-manifold
    • 5-manifold
  • Poincaré conjecture
  • List of geometric topology topics
edit  

Related portals

Portal:Algebra
Portal:Analysis
Portal:Category theory
Portal:Computer science
Portal:Cryptography
Portal:Discrete mathematics
Portal:Geometry
Algebra Analysis Category
theory
Computer
science
Cryptography Discrete
mathematics
Geometry
Portal:Logic
Portal:Mathematics
Portal:Number theory
Portal:Physics
Portal:Science
Portal:Set theory
Portal:Statistics
Portal:Topology
Logic Mathematics Number
theory
Physics Science Set theory Statistics Topology
edit  

Wikimedia

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
What are portals? · List of portals · Featured portals

Purge server cache