# Borel set of ambiguous class

$\alpha$

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))

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].