No Arabic abstract
We study the class of analytic binary relations on Polish spaces, compared with the notions of continuous reducibility or injective continuous reducibility. In particular, we characterize when a locally countable Borel relation is $Sigma$ 0 $xi$ (or $Pi$ 0 $xi$), when $xi$ $ge$ 3, by providing a concrete finite antichain basis. We give a similar characterization for arbitrary relations when $xi$ = 1. When $xi$ = 2, we provide a concrete antichain of size continuum made of locally countable Borel relations minimal among non-$Sigma$ 0 2 (or non-$Pi$ 0 2) relations. The proof of this last result allows us to strengthen a result due to Baumgartner in topological Ramsey theory on the space of rational numbers. We prove that positive results hold when $xi$ = 2 in the acyclic case. We give a general positive result in the non-necessarily locally countable case, with another suitable acyclicity assumption. We provide a concrete finite antichain basis for the class of uncountable analytic relations. Finally, we deduce from our positive results some antichain basis for graphs, of small cardinality (most of the time 1 or 2).
We establish that if it is consistent that there is a supercompact cardinal, then it is consistent that every locally compact, hereditarily normal space which does not include a perfect pre-image of omega_1 is hereditarily paracompact.
We examine locally compact normal spaces in models of form PFA(S)[S], in particular characterizing paracompact, countably tight ones as those which include no perfect pre-image of omega_1 and in which all separable closed subspaces are Lindelof.
Assume that X is a metrizable separable space, and each clopen-valued lower semicontinuous multivalued map Phi from X to Q has a continuous selection. Our main result is that in this case, X is a sigma-space. We also derive a partial converse implication, and present a reformulation of the Scheepers Conjecture in the language of continuous selections.
We prove that every usco multimap $Phi:Xto Y$ from a metrizable separable space $X$ to a GO-space $Y$ has an $F_sigma$-measurable selection. On the other hand, for the split interval $ddot{mathbb I}$ and the projection $P:ddot{mathbb I}^2to{mathbb I}^2$ of its square onto the unit square ${mathbb I}^2$, the usco multimap $P^{-1}:{mathbb I}^2multimapddot{mathbb I}^2$ has a Borel ($F_sigma$-measurable) selection if and only if the Continuum Hypothesis holds. This CH-example shows that know results on Borel selections of usco maps into fragmentable compact spaces cannot be extended to a wider class of compact spaces.
We construct, using mild combinatorial hypotheses, a real Menger set that is not Scheepers, and two real sets that are Menger in all finite powers, with a non-Menger product. By a forcing-theoretic argument, we show that the same holds in the Blass--Shelah model for arbitrary values of the ultrafilter and dominating number.