Phragmen–Brouwer theorem: Difference between revisions
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{{Short description|Equivalent properties in a normal connected locally connected topological space}} |
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In topology, the '''Phragmén–Brouwer theorem''', introduced by [[Lars Edvard Phragmén]] and [[Luitzen Egbertus Jan Brouwer]], states that if ''X'' is a [[Normal topological space|normal]] [[Connected topological space|connected]] [[locally connected topological space]], then the following two properties are equivalent: |
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*If ''A'' and ''B'' are disjoint closed subsets whose union separates ''X'', then either ''A'' or ''B'' separates ''X''. |
*If ''A'' and ''B'' are disjoint closed subsets whose union separates ''X'', then either ''A'' or ''B'' separates ''X''. |
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*''X'' is [[unicoherent]], meaning that if ''X'' is the union of two closed connected subsets, then their intersection is connected or empty. |
*''X'' is [[unicoherent]], meaning that if ''X'' is the union of two closed connected subsets, then their intersection is connected or empty. |
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The theorem remains true with the weaker condition that ''A'' and ''B'' be separated. |
The theorem remains true with the weaker condition that ''A'' and ''B'' be separated. |
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==References== |
==References== |
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*{{citation |
*{{citation | journal=[[Proceedings of the American Mathematical Society]] | volume=90 | number=2 | year=1984 | title=A Strong Form of the Phragmen–Brouwer Theorem | author=R.F. Dickman jr | pages=333–337 | doi=10.2307/2045367| jstor=2045367 }} |
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*{{citation | zbl=0337.54021 | last=Hunt | first=J.H.V. | title=The |
*{{citation | zbl=0337.54021 | last=Hunt | first=J.H.V. | title=The Phragmen–Brouwer theorem for separated sets | journal=Bol. Soc. Mat. Mex. |series=Series II | volume=19 | pages=26–35 | year=1974 }} |
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*{{Citation | last1=Wilson | first1=W. A. | title=On the Phragmén–Brouwer theorem |
*{{Citation | last1=Wilson | first1=W. A. | title=On the Phragmén–Brouwer theorem | doi=10.1090/S0002-9904-1930-04901-0 |mr=1561900 | year=1930 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=36 | issue=2 | pages=111–114| doi-access=free }} |
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* García-Maynez, A. and Illanes, A. ‘A survey of multicoherence’, An. Inst. Autonoma Mexico 29 (1989) 17–67. |
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* {{cite arXiv |last1=Brown |first1=R. |last2=Antolín-Camarena |first2=O. |title=Corrigendum to "Groupoids, the Phragmen–Brouwer Property, and the Jordan Curve Theorem", J. Homotopy and Related Structures 1 (2006) 175–183 |year=2014 |class=math.AT |eprint=1404.0556}} |
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* Wilder, R. L. Topology of manifolds, AMS Colloquium Publications, Volume 32. American Mathematical Society, New York (1949). |
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{{DEFAULTSORT:Phragmen-Brouwer theorem}} |
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[[Category:Theorems in topology]] |
[[Category:Theorems in topology]] |
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[[Category:Trees (topology)]] |
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Latest revision as of 23:43, 21 July 2022
In topology, the Phragmén–Brouwer theorem, introduced by Lars Edvard Phragmén and Luitzen Egbertus Jan Brouwer, states that if X is a normal connected locally connected topological space, then the following two properties are equivalent:
- If A and B are disjoint closed subsets whose union separates X, then either A or B separates X.
- X is unicoherent, meaning that if X is the union of two closed connected subsets, then their intersection is connected or empty.
The theorem remains true with the weaker condition that A and B be separated.
References
[edit]- R.F. Dickman jr (1984), "A Strong Form of the Phragmen–Brouwer Theorem", Proceedings of the American Mathematical Society, 90 (2): 333–337, doi:10.2307/2045367, JSTOR 2045367
- Hunt, J.H.V. (1974), "The Phragmen–Brouwer theorem for separated sets", Bol. Soc. Mat. Mex., Series II, 19: 26–35, Zbl 0337.54021
- Wilson, W. A. (1930), "On the Phragmén–Brouwer theorem", Bulletin of the American Mathematical Society, 36 (2): 111–114, doi:10.1090/S0002-9904-1930-04901-0, ISSN 0002-9904, MR 1561900
- García-Maynez, A. and Illanes, A. ‘A survey of multicoherence’, An. Inst. Autonoma Mexico 29 (1989) 17–67.
- Brown, R.; Antolín-Camarena, O. (2014). "Corrigendum to "Groupoids, the Phragmen–Brouwer Property, and the Jordan Curve Theorem", J. Homotopy and Related Structures 1 (2006) 175–183". arXiv:1404.0556 [math.AT].
- Wilder, R. L. Topology of manifolds, AMS Colloquium Publications, Volume 32. American Mathematical Society, New York (1949).