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Jordan’s curves

Authors: Fiorenza, Renato;

Jordan’s curves

Abstract

After some background, at first we prove the equivalence between two definitions of “Jordan’s curve”, defined as simple and closed plane curve but known also as subset of R2 homeomorphic to a circle. Using the ideas of Maehara ([6]) who gave a proof of Jordan’s classical theorem, for a given Jordan’s curve Γ we introduce the notions of central point, internal set, external set, all inspired by the special case in which Γ is a circle; we show that such sets are open and “arc-connected”, the first one being bounded and the second one unbounded; moreover, we show that the curve Γ is the boundary of both of them, and, finally, that they constitute a partition of R2 \ Γ. Obviously, these results include a revisitation of the proof of Jordan’s theorem. In the final section we give an explicit construction of an example of homeomorphism between a circle and the boundary of a triangle.

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
0
Average
Average
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