View source for Borel set of ambiguous class

''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017140/b0171401.png" />''

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.

====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>



====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]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Y.N. Moschovakis,   "Descriptive set theory" , North-Holland  (1980)</TD></TR></table>