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Coherent omission of intervals:\\ A combinatorial method for constructing special sets of reals and associated topological spaces\\ (invited lecture)
Last modified: 2014-03-06
Abstract
I will describe a combinatorial method for constructing subsets of the real line with strong (selective) topological properties.This method synthesizes and extendsideas from the Galvin--Miller classical construction ofa $\gamma$-set and the Bartoszy\'nski--Shelah construction of aHurewicz non-$\sigma$-compact set of reals.
This method lead to solutions of some notoriousproblems concerning the topology of the real line. Some of its major results are:
\begin{enumerate}\item A construction of a nontrivial $\gamma$-set of realsfrom a weaker hypothesis than all those hitherto used(some axiom is necessary).
\item There is a non-$\sigma$-compact, \emph{productively} Hurewicz set of reals.
\item There is (in ZFC) a set of reals, of the uncountable cardinality $\mathfrak{b}$, such that for each sequence of open covers, one can choose for each $n$ two elements $U_n$ and $V_n$ fromthe $n$-th cover, such that the sequence $U_1\cup V_1, U_2\cup V_2,\dots$ is a point-cofinite cover of our set.\\(\emph{Two} cannot be provably reduced to \emph{one} here.)
\end{enumerate}This method shows that very natural constructions produce such sets.
Credits: Item (1) is joint work with Tal Orenshtein. Item (2) is joint workwith Lyubomyr Zdomskyy. The nice name \emph{coherent omission of intervals}for the method I propose was coined by Zdomskyy.
This method lead to solutions of some notoriousproblems concerning the topology of the real line. Some of its major results are:
\begin{enumerate}\item A construction of a nontrivial $\gamma$-set of realsfrom a weaker hypothesis than all those hitherto used(some axiom is necessary).
\item There is a non-$\sigma$-compact, \emph{productively} Hurewicz set of reals.
\item There is (in ZFC) a set of reals, of the uncountable cardinality $\mathfrak{b}$, such that for each sequence of open covers, one can choose for each $n$ two elements $U_n$ and $V_n$ fromthe $n$-th cover, such that the sequence $U_1\cup V_1, U_2\cup V_2,\dots$ is a point-cofinite cover of our set.\\(\emph{Two} cannot be provably reduced to \emph{one} here.)
\end{enumerate}This method shows that very natural constructions produce such sets.
Credits: Item (1) is joint work with Tal Orenshtein. Item (2) is joint workwith Lyubomyr Zdomskyy. The nice name \emph{coherent omission of intervals}for the method I propose was coined by Zdomskyy.
Keywords
Selection Principles