Topological space

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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.


  • For a normal topological space, there exist three different approaches to define its dimension, due to Lebesgue and Čech (covering dimension dim), Urysohn and Menger (small inductive dimension ind), and Poincaré, Brouwer, and Čech (large inductive dimension Ind). All the three dimensions coincide for separable metric spaces (Tumarkin-Urysohn–Hurewitz's theorem proved in the late 20s), the equality dim X = Ind X holds for any metric space X (Katetov–Morita's theorem, 1954), while in 1962, Roy [Roy] constructed his famous example of a complete metric space Y such that ind Y = 0, but dim Y = Ind Y = 1.

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