analytic sets define a closure operator
If then .
For example, if is a collection of -analytic sets then gives and so all -analytic sets are also -analytic. In particular, for a metric space, the analytic sets are the same regardless of whether they are defined with respect to the collection of open, closed or Borel sets.
Properties 1 and 2 follow directly from the definition of analytic sets. We just need to prove 3. So, for any we show that . First, there is a compact paved space (http://planetmath.org/PavedSpace) and such that is equal to the projection . Write
for and . It is clear that is -analytic and, as countable unions and intersections of analytic sets are analytic, is also -analytic. Finally, since projections of analytic sets are analytic, must be -analytic as required.
|Title||analytic sets define a closure operator|
|Date of creation||2013-03-22 18:46:30|
|Last modified on||2013-03-22 18:46:30|
|Last modified by||gel (22282)|