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Displaying similar documents to “One-to-one continuous images of a line”

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.

Concerning the sum of a countable number of mutually exclusive continua in the plane

R. Moore (1924)

Fundamenta Mathematicae

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In 1918 Sierpiński showed that if the sum of a countably infinite collection of closed point sets is bounded then it is not a continuum. He raised the question weather this theorem remains true if the restriction that the sum should be bounded is removed from the hypothesis. The purpose of the present paper is to show that for the case where each point set of the collection in question is itself a continuum, this question may be answered in the affirmative.

Concerning continuous curves

R. Wilder (1925)

Fundamenta Mathematicae

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The present paper has three main objects: 1. to study the analogy between ordinary two-dimensional space and a plane continuous curve; 2. to characterize and analyze the boundaries of the domains complementary to a plane continuous curve; 3. to give a new characterization of continuous curves suitable for any number of dimensions;