Abstract
Develops the basic machinery of metric spaces. Includes open and closed sets, convergence of sequences, continuous functions and completeness.
I was at the mathematical school, where the master taught his pupils by a method scarce imaginable to us in Europe. The proposition and demonstration were fairly written on a thin wafer, with ink composed of a cephalic tincture. This the student was to swallow upon a fasting stomach, and for three days following eat nothing but bread and water. As the wafer digested, the tincture mounted to his brain bearing the proposition with it.
J. Swift. Gulliver’s Travels: A Voyage to Laputa.
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Notes
- 1.
Often called Cauchy’s criterion. However, it is unclear whether “Cauchy’s criterion” refers to the principle or the condition. It is an advantage to distinguish the two.
- 2.
We occasionally indulge in the entertaining sport of challenging the reader to find tacit applications of the axiom of choice.
- 3.
The idea that a set with a structure is an ordered pair is quite conventional and often convenient. Nevertheless it may be objected that it is superfluous. In the case of a measurable space \((X,{\mathcal {M}})\) the set X is the maximum element of \({\mathcal {M}}\) and does not need to be mentioned explicitly. A similar remark applies to viewing a topological space as an ordered pair \((X,{\mathcal {T}})\) where \({\mathcal {T}}\) is a topology on X.
- 4.
This follows from the well-ordering theorem, that every set can be well-ordered.
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Magnus, R. (2022). Basic Theory of Metric Spaces. In: Metric Spaces. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-030-94946-4_2
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DOI: https://doi.org/10.1007/978-3-030-94946-4_2
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