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* : → 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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