proof of equivalent definitions of analytic sets for paved spaces
for and . Rearranging this expression,
So, defining by
Setting gives the required expression, and it only remains to show that is closed. So, let be a sequence in converging to a limit . For any then for all and large enough . Hence,
showing that and that is indeed closed.
Then, for ,
Here, if , we have used the fact that is closed to deduce that for large , there is no with and, therefore, . The result of the Souslin scheme is then
(4) implies (5): Suppose that is the result of a Souslin scheme . Let us first consider the case where is Cantor space, , which is a compact Polish space. Then, for any , let be the set of such that if for some and for all other . These are closed and, therefore, compact sets.
Given any sequence it is easily seen that is nonempty if and only if there is an such that for all . Define the set in by
The projection of onto is then
which is the result of the scheme as required. The result then generalizes to any uncountable Polish space , as all such spaces contain Cantor space (http://planetmath.org/UncountablePolishSpacesContainCantorSpace).
|Title||proof of equivalent definitions of analytic sets for paved spaces|
|Date of creation||2013-03-22 18:48:36|
|Last modified on||2013-03-22 18:48:36|
|Last modified by||gel (22282)|