Difference between revisions of "Addition theorem"
From Encyclopedia of Mathematics
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| − | If a Hausdorff compactum | + | 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]]. |
====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> | <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> | ||
Revision as of 20:46, 6 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 [1]) was established in [3] for $\tau=\aleph_0$ and in [4] in complete generality. Cf. Weight of a topological space.
References
| [1] | P.S. Aleksandrov, P. Urysohn, "Mémoire sur les espaces topologiques compacts" , Koninkl. Nederl. Akad. Wetensch. , Amsterdam (1929) |
| [2] | R. Engelking, "General topology" , PWN (1977) (Translated from Polish) |
| [3] | 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) |
| [4] | 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) |
| [5] | A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian) |
How to Cite This Entry:
Addition theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Addition_theorem&oldid=26133
Addition theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Addition_theorem&oldid=26133
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