Topological geometry deals with incidence structures consisting of a point set and a family of subsets of called lines or circles etc. such that both and carry a topology and all geometric operations like joining points by a line or intersecting lines are continuous. As in the case of topological groups, many deeper results require the point space to be (locally) compact and connected. This generalizes the observation that the line joining two distinct points in the Euclidean plane depends continuously on the pair of points and the intersection point of two lines is a continuous function of these lines.
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Topologicalgeometry deals with incidence structures consisting of a point set P {\displaystyle P} and a family L {\displaystyle {\mathfrak {L}}} of subsets...
invariant under such deformations is a topological property. The following are basic examples of topological properties: the dimension, which allows...
discrete topology. With this topology, G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology...
In applied mathematics, topological data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information...
C*-algebras to usual topological spaces, the extension to noncommutative rings and algebras requires non-trivial generalization of topological spaces as "non-commutative...
parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties...
Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental...
In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time...
mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. While...
of mathematics, a topological manifold is a topological space that locally resembles real n-dimensional Euclidean space. Topological manifolds are an important...
generalizations of that. Planes are used in many areas of geometry. For instance, planes can be studied as a topological surface without reference to distances or angles;...
symplectic manifold are global in nature and topological aspects play a prominent role in symplectic geometry. The first result in symplectic topology is...
In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition...
finite union of topological curves. When complex zeros are considered, one has a complex algebraic curve, which, from the topological point of view, is...
difference with Riemannian geometry is that not every differentiable manifold need admit a symplectic form; there are certain topological restrictions. For example...
topology is homotopy theory. The study of metric spaces is geometry, the study of topological spaces is topology. The terms are not used completely consistently:...
Solomon Lefschetz, was cited in algebraic geometry to justify the use of topological techniques for algebraic geometry over any algebraically closed field K...
physics, topological order is a kind of order in the zero-temperature phase of matter (also known as quantum matter). Macroscopically, topological order...
called the vertices and the edges of the topological graph. It is usually assumed that any two edges of a topological graph cross a finite number of times...
of smooth topological invariants of such manifolds, such as de Rham cohomology or the intersection form, as well as smoothable topological constructions...
King introduced "Topological Resolution Towers" as topological models of real algebraic sets, from this they obtained new topological invariants of real...
geometries (Euclidean, spherical, or hyperbolic). In three dimensions, it is not always possible to assign a single geometry to a whole topological space...
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an...
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