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arXiv · 2312.13047

On compact subsets of the reals

Wojciech Bielas·Mateusz Kula·Szymon Plewik
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Abstract

Motivated by results of J. R. Kline and R. L. Moore (1919) that a compact subset of the plane, homeomorphic to a subset of the reals, lies on the arc, we give a purely topological characterisation of compact sets of the reals. This allows us to reduce investigations of Cantorvals to properties of countable linear orders and to show, applying the Mazurkiewicz--Sierpi\'nski Theorem (1920), that there exist continuum many non-homeomorphic L-Cantorvals.

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Wojciech Bielas, Mateusz Kula, Szymon Plewik. 2023-12-20. On compact subsets of the reals. https://arxiv.org/abs/2312.13047

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