Difference between revisions of "Totally-imperfect space"
From Encyclopedia of Mathematics
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A [[Topological space|topological space]] not containing a subset homeomorphic to a [[Cantor set|Cantor set]]. For instance, any complete separable uncountable metrizable space contains an uncountable subspace that, as well as its complement, is totally imperfect. | A [[Topological space|topological space]] not containing a subset homeomorphic to a [[Cantor set|Cantor set]]. For instance, any complete separable uncountable metrizable space contains an uncountable subspace that, as well as its complement, is totally imperfect. | ||
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====Comments==== | ====Comments==== | ||
| − | A subset of | + | A subset of $\mathbf R$ such that it and its complement are totally imperfect is usually called a Bernstein set, after F. Bernstein [[#References|[a1]]]. Such sets are non-measurable. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F. Bernstein, "Zur Theorie der trigonometrischen Reihe" ''Ber. K. Sächs. Ges. Wissenschaft. Leipzig Math.-Phys. Kl.'' , '''60''' (1908) pp. 325–338</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F. Bernstein, "Zur Theorie der trigonometrischen Reihe" ''Ber. K. Sächs. Ges. Wissenschaft. Leipzig Math.-Phys. Kl.'' , '''60''' (1908) pp. 325–338</TD></TR></table> | ||
Latest revision as of 12:02, 5 July 2014
A topological space not containing a subset homeomorphic to a Cantor set. For instance, any complete separable uncountable metrizable space contains an uncountable subspace that, as well as its complement, is totally imperfect.
Comments
A subset of $\mathbf R$ such that it and its complement are totally imperfect is usually called a Bernstein set, after F. Bernstein [a1]. Such sets are non-measurable.
References
| [a1] | F. Bernstein, "Zur Theorie der trigonometrischen Reihe" Ber. K. Sächs. Ges. Wissenschaft. Leipzig Math.-Phys. Kl. , 60 (1908) pp. 325–338 |
How to Cite This Entry:
Totally-imperfect space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Totally-imperfect_space&oldid=17317
Totally-imperfect space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Totally-imperfect_space&oldid=17317
This article was adapted from an original article by A.A. Mal'tsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article