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 {{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]]. | 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]]. | ||
Latest revision as of 15:05, 1 May 2014
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://encyclopediaofmath.org/index.php?title=Addition_theorem&oldid=32054
Addition theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Addition_theorem&oldid=32054
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