Abstract
Here we look briefly at problems to do with curves and surfaces, lengths and areas, starting with the great question of: what is a continuous curve? In particular, does a simple closed curve on a sphere have a simply-connected interior and a simply-connected exterior? Then we return to the problem of distinguishing a the unit interval from the unit square, and we conclude by showing how curves can have areas (non-zero Lebesgue measure). In this way we shall pick up some of the roots of both point-set and algebraic or geometric topology.
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- 1.
See vol. 3, 587–594. It does not seem to occur in some later reprints, e.g. in 1959.
- 2.
Non-rectifiable curves make for other problems too: What is the integral in the Cauchy integral theorem for closed curves that do not have a length, such as the von Koch snowflake curve?
- 3.
Quoted in Hales (2007, 45) from which the next few paragraphs are drawn.
- 4.
Moore also discussed continuous curves with no tangents in the same spirit.
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Gray, J. (2015). Topology. In: The Real and the Complex: A History of Analysis in the 19th Century. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-23715-2_29
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DOI: https://doi.org/10.1007/978-3-319-23715-2_29
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