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A Borel subset of a metric, or (more general) of a perfectly-normal topological, space that is at the same time a set of additive class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017140/b0171402.png" /> and of multiplicative class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017140/b0171403.png" />, i.e. belongs to the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017140/b0171404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017140/b0171405.png" /> at the same time. Borel sets of ambiguous class 0 are the closed and open sets. Borel sets of ambiguous class 1 are sets of types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017140/b0171406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017140/b0171407.png" /> at the same time. Any Borel set of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017140/b0171408.png" /> is a Borel set of ambiguous class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017140/b0171409.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017140/b01714010.png" />. The Borel sets of ambiguous class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017140/b01714011.png" /> form a field of sets.
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A [[Borel set|Borel subset]] of a [[metric space]], or more generallly, a [[Perfectly-normal space|perfectly-normal topological space]], that is at the same time a set of additive class $\alpha$ and of multiplicative class $\alpha$, i.e. belongs to the classes $F_\alpha$ and $G_\alpha$ at the same time. The Borel sets of ambiguous class 0 are the [[Open-closed set|closed and open set]]s. Borel sets of ambiguous class 1 are sets of types [[F-sigma|$F_\sigma$]] and [[G-delta|$G_\delta$]] at the same time. Any Borel set of class $\alpha$ is a Borel set of ambiguous class $\beta$ for any $\beta > \alpha$. The Borel sets of ambiguous class $\alpha$ form a field of sets.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Kuratowski,  "Topology" , '''1''' , Acad. Press  (1966)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Hausdorff,  "Grundzüge der Mengenlehre" , Leipzig  (1914)  (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  K. Kuratowski,  "Topology" , '''1''' , Acad. Press  (1966)  (Translated from French)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  F. Hausdorff,  "Grundzüge der Mengenlehre" , Leipzig  (1914)  (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))</TD></TR>
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</table>
  
  
  
 
====Comments====
 
====Comments====
The notations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017140/b01714012.png" /> are still current in topology. Outside topology one more often uses the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017140/b01714013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017140/b01714014.png" />, respectively. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017140/b01714015.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017140/b01714016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017140/b01714017.png" />; but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017140/b01714018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017140/b01714019.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017140/b01714020.png" />. The notation for the ambiguous classes is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017140/b01714021.png" />. See also [[#References|[a1]]].
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The notations $F_\alpha$, $G_\alpha$ are still current in topology. Outside topology one more often uses the notation $\Sigma^0_\alpha$, $\Pi^0_\alpha$, respectively. For $\alpha \ge  \omega$ one has $F_\alpha = \Sigma^0_\alpha$, $G_\alpha = \Pi^0_\alpha$; but for $n < \omega$ one has $F_n = \Sigma^0_{n+1}$ and $G_n = \Pi^0_{n+1}$. The notation for the ambiguous classes is $\Delta^0_\alpha = \Sigma^0_\alpha \cap \Pi^0_\alpha$. See also [[#References|[a1]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Y.N. Moschovakis,  "Descriptive set theory" , North-Holland  (1980)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  Y.N. Moschovakis,  "Descriptive set theory" , North-Holland  (1980)</TD></TR>
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</table>
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Latest revision as of 20:49, 20 November 2016

$\alpha$

A Borel subset of a metric space, or more generallly, a perfectly-normal topological space, that is at the same time a set of additive class $\alpha$ and of multiplicative class $\alpha$, i.e. belongs to the classes $F_\alpha$ and $G_\alpha$ at the same time. The Borel sets of ambiguous class 0 are the closed and open sets. Borel sets of ambiguous class 1 are sets of types $F_\sigma$ and $G_\delta$ at the same time. Any Borel set of class $\alpha$ is a Borel set of ambiguous class $\beta$ for any $\beta > \alpha$. The Borel sets of ambiguous class $\alpha$ form a field of sets.

References

[1] K. Kuratowski, "Topology" , 1 , Acad. Press (1966) (Translated from French)
[2] F. Hausdorff, "Grundzüge der Mengenlehre" , Leipzig (1914) (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))


Comments

The notations $F_\alpha$, $G_\alpha$ are still current in topology. Outside topology one more often uses the notation $\Sigma^0_\alpha$, $\Pi^0_\alpha$, respectively. For $\alpha \ge \omega$ one has $F_\alpha = \Sigma^0_\alpha$, $G_\alpha = \Pi^0_\alpha$; but for $n < \omega$ one has $F_n = \Sigma^0_{n+1}$ and $G_n = \Pi^0_{n+1}$. The notation for the ambiguous classes is $\Delta^0_\alpha = \Sigma^0_\alpha \cap \Pi^0_\alpha$. See also [a1].

References

[a1] Y.N. Moschovakis, "Descriptive set theory" , North-Holland (1980)
How to Cite This Entry:
Borel set of ambiguous class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borel_set_of_ambiguous_class&oldid=39789
This article was adapted from an original article by A.G. El'kin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article