Topological space

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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 formalizin

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"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–Hurewitzs theorem proved in the late 20s), the equality dim X = Ind X holds for any metric space X (Katetov–Moritas 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."

Topological space