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An outside view on the Jordan curve theorem
by
Vladimir Todorov
University of Architecture, Civil Engineering an Geodesy, Bulgaria
Let \Sigma = { (F-i, F+i ) | i = 1, ... n } be a family of disjoint pairs of closed subsets of the topological space X. \Sigma is said to be an essential family if for an arbitrary collection { Ci }i=1n, of separators Ci between F-i and F+i in X we have \cap i=1n Ci =/= \emptyset. We will say that \Sigma is minimal if it is minimal by inclusion. We define a frame of \Sigma as the sum of the sets F-i and F+i.
The compact set K in Rn is said to be an irreducible boundary if Rn \K is disconnected and K is a boundary of each component of Rn \K. This paper contains a necessary and sufficient condition which describes irreducible boundaries in Rn: the compact K subset Rn is an irreducible boundary if and only if it is a frame of some minimal essential family. As a corollary we obtain a simple proof of the classical Jordan curve theorem.
Date received: February 1, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cady-26.