# 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)

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.