Namespaces
Variants
Actions

Difference between revisions of "Totally-imperfect space"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
 +
{{TEX|done}}
 
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.
  
Line 4: Line 5:
  
 
====Comments====
 
====Comments====
A subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093460/t0934601.png" /> 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.
+
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=32380
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