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Difference between revisions of "Addition theorem"

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If a Hausdorff compactum $X$ can be represented as the union over a set of infinite cardinality $\leq\tau$ of its subspaces of weight $\leq\tau$, then the weight of $X$ does not exceed $\tau$. The addition theorem (which was formulated as a problem in [[#References|[1]]]) was established in [[#References|[3]]] for $\tau=\aleph_0$ and in [[#References|[4]]] in complete generality. Cf. [[Weight of a topological space|Weight of a topological space]].
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If a Hausdorff compactum $X$ can be represented as the union over a set of infinite cardinality $\leq\tau$ of its subspaces of weight $\leq\tau$, then the weight of $X$ does not exceed $\tau$. The addition theorem (which was formulated as a problem in {{Cite|AlUr}}) was established in {{Cite|Sm}} for $\tau=\aleph_0$ and in {{Cite|Ar}} in complete generality. Cf. [[Weight of a topological space|Weight of a topological space]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.S. Aleksandrov,  P. Urysohn,  "Mémoire sur les espaces topologiques compacts" , Koninkl. Nederl. Akad. Wetensch. , Amsterdam  (1929)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Engelking,  "General topology" , PWN  (1977)  (Translated from Polish)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Yu.M. Smirnov,  "On metrizability of bicompacta, decomposable as a sum of sets with a countable base"  ''Fund. Math.'' , '''43'''  (1956)  pp. 387–393  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.V. Arkhangel'skii,  "An addition theorem for weights of sets lying in bicompacta"  ''Dokl. Akad. Nauk SSSR'' , '''126''' :  2  (1959)  pp. 239–241  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A.V. Arkhangel'skii,  V.I. Ponomarev,  "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR></table>
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|valign="top"|{{Ref|AlUr}}||valign="top"| P.S. Aleksandrov,  P. Urysohn,  "Mémoire sur les espaces topologiques compacts", Koninkl. Nederl. Akad. Wetensch., Amsterdam  (1929)
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|valign="top"|{{Ref|Ar}}||valign="top"| A.V. Arkhangel'skii,  "An addition theorem for weights of sets lying in bicompacta"  ''Dokl. Akad. Nauk SSSR'', '''126''' :  2  (1959)  pp. 239–241  (In Russian)
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|valign="top"|{{Ref|ArPo}}||valign="top"| A.V. Arkhangel'skii,  V.I. Ponomarev,  "Fundamentals of general topology: problems and exercises", Reidel  (1984)  (Translated from Russian)
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|valign="top"|{{Ref|En}}||valign="top"|  R. Engelking,  "General topology", PWN  (1977)  (Translated from Polish)
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|valign="top"|{{Ref|Sm}}||valign="top"|  Yu.M. Smirnov,  "On metrizability of bicompacta, decomposable as a sum of sets with a countable base"  ''Fund. Math.'', '''43'''  (1956)  pp. 387–393  (In Russian)
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Revision as of 01:04, 7 May 2012

for weights


If a Hausdorff compactum $X$ can be represented as the union over a set of infinite cardinality $\leq\tau$ of its subspaces of weight $\leq\tau$, then the weight of $X$ does not exceed $\tau$. The addition theorem (which was formulated as a problem in [AlUr]) was established in [Sm] for $\tau=\aleph_0$ and in [Ar] in complete generality. Cf. Weight of a topological space.

References

[AlUr] P.S. Aleksandrov, P. Urysohn, "Mémoire sur les espaces topologiques compacts", Koninkl. Nederl. Akad. Wetensch., Amsterdam (1929)
[Ar] A.V. Arkhangel'skii, "An addition theorem for weights of sets lying in bicompacta" Dokl. Akad. Nauk SSSR, 126 : 2 (1959) pp. 239–241 (In Russian)
[ArPo] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises", Reidel (1984) (Translated from Russian)
[En] R. Engelking, "General topology", PWN (1977) (Translated from Polish)
[Sm] Yu.M. Smirnov, "On metrizability of bicompacta, decomposable as a sum of sets with a countable base" Fund. Math., 43 (1956) pp. 387–393 (In Russian)
How to Cite This Entry:
Addition theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Addition_theorem&oldid=26134
This article was adapted from an original article by A.V. Arkhangel'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article