A Taste of Topology pp 23-60 | Cite as

# Metric Spaces

## Abstract

What is the minimum of structure one needs to have on a set in order to be able to speak of continuity?

If *f* is a function defined on a subset of R—or, more generally, of Euclidean *n*-space R^{ n }—we say that *f* is continuous at *x*_{0} if “*f(x)* approaches *f(x*_{ 0 }*)* as *x* approaches *x*_{ 0 }.” With *ε* and *δ*, this statement can be made sufficiently precise for mathematical purposes.

*For each ε* > 0, *there is δ* > 0 *such that* |*f(x) - f(x*_{ 0 }*)*| < *ε for all x such that* |*x - x*_{ 0 }| < *δ*.

Crucial for the definition of continuity thus seems to be that we can measure the distance between two real numbers (or, rather, two vectors in Euclidean *n*-space).

If we want to speak of continuity of functions defined on more general sets, we should thus have a meaningful way to speak of the distance between two points of such a set: this, in a nutshell, is the idea behind a metric space.

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