Topology (structure)
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 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'Huillier, is at the origin of topology. 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."
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