Portal:Topology
Topology
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, that satisfy certain properties, turning the given set into what is known as a topological space. Important topological properties include connectedness and compactness.
Topology developed as a field of study out of geometry and set theory, through analysis of concepts such as space, dimension, and transformation. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs (Greek-Latin for "geometry of place") and analysis situs (Greek-Latin for "picking apart of place"). Leonhard Euler's Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics.
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... |
WikiProjects
The Mathematics WikiProject is the center for mathematics-related editing on Wikipedia. Join the discussion on the project's talk page.
Project pages
Essays
Subprojects
Related projects
Selected image
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... |
Did you know?
- ...that it is possible to turn a sphere inside out without tears or creases? (The sphere is allowed to pass through itself).
- ...that the Klein bottle gives a two-fold covering space of itself?
- ...that outstanding mathematician Grigori Perelman was offered a Fields Medal in 2006, in part for his proof of the Poincaré conjecture, which he declined?
- ...that you cannot knot strings in 4-dimensions? You can, however, knot 2-dimensional surfaces like spheres.
Categories
Topics in Topology
Main articles | Key concepts | Algebraic topology | Geometric topology |
---|---|---|---|
Related portals
Algebra | Analysis |
Category theory |
Computer science |
Cryptography |
Discrete mathematics |
Logic | Mathematics |
Number theory |
Physics | Science | Set theory | Statistics |