Topology by Richard Earl

Author:Richard Earl [Earl, Richard]
Language: eng
Format: epub
ISBN: 9780192568991
Publisher: OUP Oxford
Published: 2019-11-15T00:00:00+00:00

Thoughts on distance

Consider the important notions needed to define continuity in Chapter 3. Loosely put, continuity requires that we can constrain the difference in outputs by suitably constraining the difference in inputs. We will need a more general notion of the difference between other types of input and output: for example, what is the ‘difference’ between two points of the plane?

Given two real numbers, x and y, we denote the difference between them as |x–y|, and this is just another expression for the distance between x and y as points on the real line (Figure 20(b)). Likewise, given two points (x1, y1) and (x2, y2) in the plane, we might take the ‘difference’ between them to mean the straight-line distance between them (Figure 28(a)). Pythagoras’ Theorem tells us this distance equals

28. Visualizing different metrics (a) Straight line distance, (b) Taxicab distance, (c) Distance between functions.

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