The main result is the following. Let f:X→Y be a continuous mapping of a completely Baire space X onto a hereditary weakly Preiss-Simon regular space Y such that the image of every open subset of X ...is a resolvable set in Y. Then Y is completely Baire.
The classical Hurewicz theorem about closed embedding of the space of rational numbers into metrizable spaces is generalized to weakly Preiss-Simon regular spaces.
Non-meager free sets and independent families MEDINI, ANDREA; REPOVŠ, DUŠAN; ZDOMSKYY, LYUBOMYR
Proceedings of the American Mathematical Society,
09/2017, Volume:
145, Issue:
9
Journal Article
Peer reviewed
Open access
Our main result is that, given a collection \mathcal {R} of meager relations on a Polish space X such that \vert\mathcal {R}\vert\leq \omega , there exists a dense Baire subspace F of X ...(equivalently, a nowhere meager subset F of X) such that F is R-free for every R\in \mathcal {R}. This generalizes a recent result of Banakh and Zdomskyy. As an application, we show that there exists a non-meager independent family on \omega , and define the corresponding cardinal invariant. Furthermore, assuming Martin's Axiom for countable posets, our result can be strengthened by substituting `` \vert\mathcal {R}\vert\leq \omega '' with `` \vert\mathcal {R}\vert<\mathfrak{c}'' and ``Baire'' with ``completely Baire''.
f:X→Y is an Fσ-mapping if f maps Fσ-sets in X to Fσ-sets in Y and f−1 maps Fσ-sets in Y back to Fσ-sets in X. R.W. Hansell, J.E. Jayne, and C.A. Rogers under some assumptions proved that every ...Fσ-mapping of an absolute Souslin-F set onto an absolute Souslin-F set is piecewise closed. We give a similar result for Souslin-F subsets of completely Baire spaces.
W. Hurewicz proved that analytic Menger sets of reals are σ-compact and that co-analytic completely Baire sets of reals are completely metrizable. It is natural to try to generalize these theorems to ...projective sets. This has previously been accomplished by V=L for projective counterexamples, and the Axiom of Projective Determinacy for positive results. For the first problem, the first author, S. Todorcevic, and S. Tokgöz have produced a finer analysis with much weaker axioms. We produce a similar analysis for the second problem, showing the two problems are essentially equivalent. We also construct in ZFC a separable metrizable space with ωth power completely Baire, yet lacking a dense completely metrizable subspace. This answers a question of Eagle and Tall in Abstract Model Theory.
On the Jayne–Rogers theorem Medvedev, S. V.
Acta mathematica Hungarica,
08/2018, Volume:
155, Issue:
2
Journal Article
Peer reviewed
J. E. Jayne and C. A. Rogers
3
proved that a mapping
f
:
X
→
Y
of an absolute Souslin-
F
set
X
to a metric space
Y
is
Δ
2
0
-measurable if and only if it is piecewise continuous. We give a similar ...result for a perfectly paracompact first-countable space
X
and a regular space
Y
.
Between Polish and completely Baire Medini, Andrea; Zdomskyy, Lyubomyr
Archive for mathematical logic,
02/2015, Volume:
54, Issue:
1-2
Journal Article
Peer reviewed
Open access
All spaces are assumed to be separable and metrizable. Consider the following properties of a space
X
.
X
is Polish.
For every countable crowded
Q
⊆
X
there exists a crowded
Q
′
⊆
Q
with compact ...closure.
Every closed subspace of
X
is either scattered or it contains a homeomorphic copy of
2
ω
.
Every closed subspace of
X
is a Baire space.
While (4) is the well-known property of being
completely Baire
, properties (2) and (3) have been recently introduced by Kunen, Medini and Zdomskyy, who named them the
Miller property
and the
Cantor-Bendixson property
respectively. It turns out that the implications
(
1
)
→
(
2
)
→
(
3
)
→
(
4
)
hold for every space
X
. Furthermore, it follows from a classical result of Hurewicz that all these implications are equivalences if
X
is coanalytic. Under the axiom of Projective Determinacy, this equivalence result extends to all projective spaces. We will complete the picture by giving a
ZFC
counterexample and a consistent definable counterexample of lowest possible complexity to the implication
(
i
)
←
(
i
+
1
)
for
i
=
1
,
2
,
3
. For one of these counterexamples we will need a classical theorem of Martin and Solovay, of which we give a new proof, based on a result of Baldwin and Beaudoin. Finally, using a method of Fischer and Friedman, we will investigate how changing the value of the continuum affects the definability of these counterexamples. Along the way, we will show that every uncountable completely Baire space has size continuum.
Using the property of being completely Baire, countable dense homogeneity and the perfect set property we will be able, under Martinʼs Axiom for countable posets, to distinguish non-principal ...ultrafilters on ω up to homeomorphism. Here, we identify ultrafilters with subpaces of 2ω in the obvious way. Using the same methods, still under Martinʼs Axiom for countable posets, we will construct a non-principal ultrafilter U⊆2ω such that Uω is countable dense homogeneous. This consistently answers a question of Hrušák and Zamora Avilés. Finally, we will give some partial results about the relation of such topological properties with the combinatorial property of being a P-point.
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