Borel set of ambiguous class

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 and of multiplicative class , i.e. belongs to the classes and 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 and at the same time. Any Borel set of class is a Borel set of ambiguous class for any . The Borel sets of ambiguous class 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 are still current in topology. Outside topology one more often uses the notation , , respectively. For one has , ; but and for . The notation for the ambiguous classes is . See also [a1].

References

[a1]	Y.N. Moschovakis, "Descriptive set theory" , North-Holland (1980)