Universally meager sets

Abstract

Abstract. We study category counterparts of the notion of a universal measure zero set of reals.We say that a set A ⊆ R is universally meager if every Borel isomorphic image of A is meager in R. We give various equivalent definitions emphasizing analogies with the universally null sets of reals.In particular, two problems emerging from an earlier work of Grzegorek are solved.

Countably Perfectly Meager Sets

“…But then E is a homeomorphic copy of H in 2 N which is not PM in 2 N . We close this section by showing that like UM (see [24]) (but, at least consistencywise, unlike PM; see [19]), the class PM is closed under products.…”

“…Under the notation from Theorem 1.1 it follows that if T is universally meager then H is universally meager as well (UM being closed with respect to preimages under continuous injections, see [24]) and so is its homeomorphic copy E. A refinement of this argument also shows that, at least consistently, in contrast to both PM and UM the class PM is not closed with respect to homeomorphic images (see Theorem 3.1(3)). Consequently, unlike in the case of PM and UM, the statement that a subspace A of a Polish space is countably perfectly meager makes sense only if we specify a Polish space X in which it is embedded.…”

“…It suffices to notice that if T is either universally meager or a -set, then so is E, respectively (cf. [24]). But E ∈ PM .…”

“…Let us recall that a subset A of a perfect Polish space X is universally meager (A ∈ UM; see [1,2,24,25]), if for every Borel isomorphism f between X and any perfect Polish space Y the image of A under f is meager in Y (this class of sets was earlier introduced and studied by Grzegorek [8][9][10] under the name of absolutely of the first category sets).…”

On a Construction of Universally Null Sets

“…X ⊂ R n is perfectly meager if X ∩ D is meager in D for each perfect set D ⊂ R n . E. Grzegorek (see [1]) defined a bit stronger (at least under CH) property which was proved to be equivalent to the following one (see [5]). X ⊂ R n is universally meager (universally Baire) if f [X] is meager (has Baire Property) for each Borel isomorphism f : R n → R.…”