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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.90 Topological Spaces

Ben Green

A topological space is the most basic context in which one can understand the notion of a CONTINUOUS FUNCTION [I.3 §5.2].

Let us recall a standard definition of what it means for a function f : ImageImage to be continuous. Suppose that f (x) = y. Then f is continuous at x provided that f (x′) is close to y whenever x′ is close to x. Of course, to make this a mathematically rigorous notion we have to be precise about the meaning of “close.” We could say that f (x′) is close to y if | f (x′) – f (x)| < ε, where ε > 0 ...

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