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A three-dimensional model of a figure-eight knot.The figure-eight knot is a prime knot and has an Alexander–Briggs notation of 4 1.. Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling ...
The topography of an area may refer to the landforms and features themselves, or a description or depiction in maps. Topography is a field of geoscience and planetary science and is concerned with local detail in general, including not only relief, but also natural, artificial, and cultural features such as roads, land boundaries, and buildings ...
Examples of different knots including the trivial knot (top left) and the trefoil knot (below it) A knot diagram of the trefoil knot, the simplest non-trivial knot. In topology, knot theory is the study of mathematical knots.
In mathematics, geometry and topology is an umbrella term for the historically distinct disciplines of geometry and topology, as general frameworks allow both disciplines to be manipulated uniformly, most visibly in local to global theorems in Riemannian geometry, and results like the Gauss–Bonnet theorem and Chern–Weil theory.
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance.More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms ...
Examples of closed surfaces include the sphere, the torus and the Klein bottle. Examples of non-closed surfaces include an open disk (which is a sphere with a puncture), a cylinder (which is a sphere with two punctures), and the Möbius strip. A surface embedded in three-dimensional space is closed if and only if it is the boundary of a solid.
The Topologist's sine curve, a useful example in point-set topology.It is connected but not path-connected. In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology.
An important example of this programmatic idea is the étale topos of a scheme. Another illustration of the capability of Grothendieck topoi to incarnate the “essence” of different mathematical situations is given by their use as "bridges" for connecting theories which, albeit written in possibly very different languages, share a common ...