Calculus Small Changes, Big Ideas

Section 21.4 Topology of the real numbers

• A metric on a set \(X\) is a function \(d:X\times X\to\bb R\) that satisfies the following properties:

  1. Positive definite: \(d(x,y)\ge 0\) for all \(x,y\in X\text{,}\) and \(d(x,y)=0\) if and only if \(x=y\text{.}\)

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  2. Symmetric: \(d(x,y)=d(y,x)\) for all \(x,y\in X\text{.}\)

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  3. Triangle inequality: \(d(x,y)+d(y,z)\ge d(x,z)\) for all \(x,y,z\in X\text{.}\)

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• The absolute value can be used as a distance function: \(d(x,y)=|x-y|\text{.}\)

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• Other corollaries of the triangle inequality:

  • \(\displaystyle |x+y|\le |x|+|y|\)

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  • \(\displaystyle \big||x|-|y|\big|\le|x-y|\)

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  • Extended: \(\big||x|-|y|\big|\le|x\pm y|\le |x|+|y|\)

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• A set \(U\subseteq\bb R\) is open if, for every \(x\in U\text{,}\) there exists \(\delta >0\) such that \((x-\delta,x+\delta)\subseteq U\text{.}\) We call \((x-\delta,x+\delta)\) the \(\delta\)-neighborhood of \(x\) and denote it \(V_{\delta}(x)\text{,}\) so \(V_{\delta}(x)\subseteq U\text{.}\)

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• Open sets are closed under arbitrary units and finite intersections.

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• Every open set is a countable union of disjoint open intervals.

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• A set is closed if its complement is open.

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• De Morgan’s laws for open and closed sets

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