Namespaces
Variants
Actions

CA-set

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.


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.

How to Cite This Entry:
CA-set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=CA-set&oldid=46182
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article