View source for Borel set of ambiguous class

''$\alpha$''

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====
<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 $F_\alpha$, $G_\alpha$ 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====
<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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