Displaying similar documents to “Concerning the sum of a countable number of mutually exclusive continua in the plane”

Concerning the common boundary of two domains

R. Moore (1924)

Fundamenta Mathematicae

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The main purpose of the present paper is to show that if a bounded continuum has more then one prime part and no one of its prime parts separates the plane then in order that it should have just two complementary domains and be the complete boundary of each of them it is necessary and sufficient that it should remain connected in the weak sense on the removal of any one of its connected proper subsets which is closed.

The Hausdorff lower semicontinuous envelope of the length in the plane

Raphaël Cerf (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We study the Hausdorff lower semicontinuous envelope of the length in the plane. This envelope is taken with respect to the Hausdorff metric on the space of the continua. The resulting quantity appeared naturally as the rate function of a large deviation principle in a statistical mechanics context and seems to deserve further analysis. We provide basic simple results which parallel those available for the perimeter of Caccioppoli and De Giorgi.

A theorem on continua

R. Wilder (1925)

Fundamenta Mathematicae

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The purpose of this paper is to prove Theoreme: Of two concentric circles C_1 and C_2, let C_1 be the smaller. Denote by H the point set which is the sum of C_1, C_2, and the annular domain bounded by C_1 and C_2. Let M be a continuum which contains a point A interior to C_1 and a point B exterior to C_2. If N is any connected subset of M containing A and B, N will contain at least one point of some continuum which is a subset of M and H, and which has at least one point in common with...