CA-set
The complement of an -set in a complete separable metric space
; that is,
is a
-set if
is an
-set, or, in other words a
-set is a projective set of class 2. There is an example of a
-set that is not an
-set. Any
-set is a one-to-one continuous image of some
-set (Mazurkiewicz's theorem).
A point is called a value of order
of a mapping
if there is one and only one point such that
. The values of order 1 of a
-measurable mapping
on an arbitrary Borel set form a
-set (Luzin's theorem). The converse is true: Let
be any
-set belonging to a space
. Then there is a continuous function
defined on a closed subset of the irrational numbers such that
is the set of points of order 1 of
. Kuratowski's reduction theorem: Given an infinite sequence of
-sets
there is a sequence of disjoint
-sets
such that
and
.
References
[1] | K. Kuratowski, "Topology" , 1 , Acad. Press (1966) (Translated from French) |
Comments
A -set is also called a co-analytic set, their class is nowadays denoted by
. See also
-set.
CA-set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=CA-set&oldid=46182