Abstract
The notion of polygonal connectedness is introduced. It is shown to be an equivalence relation. Convex sets are shown to be connected. The equivalence of connectedness with the non-existence of discretely valued non-constant continuous functions is shown. An elementary proof of the Jordan Closed-Curve theorem is given.
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Notes
- 1.
A subset \(U\) is open if every point in \(U\) is surrounded by points in \(U.\) More formally, if \(x \in U\) there exists \(\delta >0\) such that \(N_{\delta }(x) = \{ y \, : \, |y-x| < \delta \} \subseteq U.\)
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© 2015 Springer International Publishing Switzerland
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Joshi, M. (2015). Connectedness and the Jordan Curve Theorem. In: Proof Patterns. Springer, Cham. https://doi.org/10.1007/978-3-319-16250-8_14
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DOI: https://doi.org/10.1007/978-3-319-16250-8_14
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-16249-2
Online ISBN: 978-3-319-16250-8
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