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

John Kline

Fundamenta Mathematicae (1924)

  • Volume: 5, Issue: 1, page 3-10
  • ISSN: 0016-2736

Abstract

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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 has O and only O in common with J. Theorem: Suppose M is an unbounded closed connected set such that if g is any bounded connected subset of M, then M-g is connected. Then M is not a continuous curve.

How to cite

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Kline, John. "Closed connected sets which remain connected upon the removal of certain, connected subsets." Fundamenta Mathematicae 5.1 (1924): 3-10. <http://eudml.org/doc/213936>.

@article{Kline1924,
abstract = {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 has O and only O in common with J. Theorem: Suppose M is an unbounded closed connected set such that if g is any bounded connected subset of M, then M-g is connected. Then M is not a continuous curve.},
author = {Kline, John},
journal = {Fundamenta Mathematicae},
keywords = {zbiór spójny; krzywa Jordana; krzywa zamknięta; krzywa; podzbiór właściwy; topologia},
language = {eng},
number = {1},
pages = {3-10},
title = {Closed connected sets which remain connected upon the removal of certain, connected subsets},
url = {http://eudml.org/doc/213936},
volume = {5},
year = {1924},
}

TY - JOUR
AU - Kline, John
TI - Closed connected sets which remain connected upon the removal of certain, connected subsets
JO - Fundamenta Mathematicae
PY - 1924
VL - 5
IS - 1
SP - 3
EP - 10
AB - 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 has O and only O in common with J. Theorem: Suppose M is an unbounded closed connected set such that if g is any bounded connected subset of M, then M-g is connected. Then M is not a continuous curve.
LA - eng
KW - zbiór spójny; krzywa Jordana; krzywa zamknięta; krzywa; podzbiór właściwy; topologia
UR - http://eudml.org/doc/213936
ER -

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