Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology. In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology. Around 1735, Euler discovered the formula V − E + F = 2 {displaystyle V-E+F=2} relating the number of vertices, edges and faces of a convex polyhedron, and hence of a planar graph. The study and generalization of this formula, specifically by Cauchy and L'Huilier, is at the origin of topology. In 1827, Carl Friedrich Gauss published General investigations of curved surfaces which in section 3 defines the curved surface in a similar manner to the modern topological understanding: 'A curved surface is said to possess continuous curvature at one of its points A, if the direction of all the straight lines drawn from A to points of the surface at an infinitely small distance from A are deflected infinitely little from one and the same plane passing through A.' Yet, 'until Riemann’s work in the early 1850s, surfaces were always dealt with from a local point of view (as parametric surfaces) and topological issues were never considered.' 'Möbius and Jordan seem to be the first to realize that the main problem about the topology of (compact) surfaces is to find invariants (preferably numerical) to decide the equivalence of surfaces, that is, to decide whether two surfaces are homeomorphic or not.' The subject is clearly defined by Felix Klein in his 'Erlangen Program' (1872): the geometry invariants of arbitrary continuous transformation, a kind of geometry. The term 'topology' was introduced by Johann Benedict Listing in 1847, although he had used the term in correspondence some years earlier instead of previously used 'Analysis situs'. The foundation of this science, for a space of any dimension, was created by Poincaré. His first article on this topic appeared in 1894. In the 1930s, James Waddell Alexander II and Hassler Whitney first expressed the idea that a surface is a topological space that is locally like a Euclidean plane. The utility of the notion of a topology is shown by the fact that there are several equivalent definitions of this structure. Thus one chooses the axiomatisation suited for the application. The most commonly used is that in terms of open sets, but perhaps more intuitive is that in terms of neighbourhoods and so this is given first. This axiomatization is due to Felix Hausdorff.Let X be a set; the elements of X are usually called points, though they can be any mathematical object. We allow X to be empty. Let N be a function assigning to each x (point) in X a non-empty collection N(x) of subsets of X. The elements of N(x) will be called neighbourhoods of x with respect to N (or, simply, neighbourhoods of x). The function N is called a neighbourhood topology if the axioms below are satisfied; and then X with N is called a topological space. The first three axioms for neighbourhoods have a clear meaning. The fourth axiom has a very important use in the structure of the theory, that of linking together the neighbourhoods of different points of X. A standard example of such a system of neighbourhoods is for the real line R, where a subset N of R is defined to be a neighbourhood of a real number x if it includes an open interval containing x. Given such a structure, a subset U of X is defined to be open if U is a neighbourhood of all points in U. The open sets then satisfy the axioms given below. Conversely, when given the open sets of a topological space, the neighbourhoods satisfying the above axioms can be recovered by defining N to be a neighbourhood of x if N includes an open set U such that x ∈ U.