CA-set

The complement of an $ {\mathcal A} $- set in a complete separable metric space $ X $; that is, $ P \subset X $ is a $ C {\mathcal A} $- set if $ X \setminus P $ is an $ {\mathcal A} $- set, or, in other words a $ C {\mathcal A} $- set is a projective set of class 2. There is an example of a $ C {\mathcal A} $- set that is not an $ {\mathcal A} $- set. Any $ {\mathcal A} $- set is a one-to-one continuous image of some $ C {\mathcal A} $- set (Mazurkiewicz's theorem).

A point $ y $ is called a value of order $ 1 $ of a mapping $ f $ if there is one and only one point such that $ y = f (x) $. The values of order 1 of a $ B $- measurable mapping $ f $ on an arbitrary Borel set form a $ C {\mathcal A} $- set (Luzin's theorem). The converse is true: Let $ C $ be any $ C {\mathcal A} $- set belonging to a space $ X $. Then there is a continuous function $ f $ defined on a closed subset of the irrational numbers such that $ C $ is the set of points of order 1 of $ f $. Kuratowski's reduction theorem: Given an infinite sequence of $ C {\mathcal A} $- sets $ U ^ {1} , U ^ {2} \dots $ there is a sequence of disjoint $ C {\mathcal A} $- sets $ V ^ {1} , V ^ {2} \dots $ such that $ V ^ {n} \subset U ^ {n} $ and $ \cup _ {n=1} ^ \infty V ^ {n} = \cup _ {n=1} ^ \infty U ^ {n} $.

References

[1]	K. Kuratowski, "Topology" , 1 , Acad. Press (1966) (Translated from French)

Comments

A $ C {\mathcal A} $- set is also called a co-analytic set, their class is nowadays denoted by $ \Pi _ {1} ^ {1} $. See also $ {\mathcal A} $- set.