Displaying similar documents to “A separation theorem”

Concerning the common boundary of two domains

R. Moore (1924)

Fundamenta Mathematicae

Similarity:

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.

Closed connected sets which remain connected upon the removal of certain, connected subsets

John Kline (1924)

Fundamenta Mathematicae

Similarity:

The purpose of this paper is to prove: Theorem: Suppose M is a closed connected set containing more than one point such that if g is any connected subset of M, then M-g is connected. Under these conditions M is a simple closed curve. Theorem: If M is an unbounded closed connected set which remains connected upon the removal of any unbounded connected proper subset, then M is either an open curve, a ray of an open curve or a simple closed curve J plus OP, a ray of an open curve which...

Concerning connectedness im kleinen and a related property

R. Moore (1922)

Fundamenta Mathematicae

Similarity:

Sierpinski has shown (Wacław Sierpiński Sur une condition pour qu'un continu soit une courbe jordanienne, Fundamenta Mathematicae I (1920), pp. 44-60) that in order that a closed and connected set of points M should be a continuous curve it is necessary and sufficient that, for every positive number ϵ, the connected point-set M should be the sum of a finite number of closed and connected point-sets each of diameter less than ϵ. It follows that, as applied to point-sets which are closed,...

Concerning continuous curves

R. Wilder (1925)

Fundamenta Mathematicae

Similarity:

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;

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

R. Moore (1924)

Fundamenta Mathematicae

Similarity:

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.

Removing sets from connected product spaces while preserving connectedness

Melvin Henriksen, Amir Nikou (2007)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

As per the title, the nature of sets that can be removed from a product of more than one connected, arcwise connected, or point arcwise connected spaces while preserving the appropriate kind of connectedness is studied. This can depend on the cardinality of the set being removed or sometimes just on the cardinality of what is removed from one or two factor spaces. Sometimes it can depend on topological properties of the set being removed or its trace on various factor spaces. Some of...