Piotr Szewczak
Piotr Szewczak: Perfectly meager sets in the transitive sense and the Hurewicz property
18/03/24 11:11
Tuesday, March 19, 2024 17:15
Location: A.4.1 C-19
Speaker: Piotr Szewczak (UKSW)
Title: Perfectly meager sets in the transitive sense and the Hurewicz property
Abstract: We work in the Cantor space with the usual group operation +. A set X is perfectly meager in the transitive sense if for any perfect set P there is an F-sigma set F containing X such that for every point t the intersection of t+F and P is meager in the relative topology of P. A set X is Hurewicz if for any sequence of increasing open covers of X one can select one set from each cover such that the chosen sets formulate a gamma-cover of X, i.e., an infinite cover such that each point from X belongs to all but finitely many sets from the cover. Nowik proved that each Hurewicz set which cannot be mapped continuously onto the Cantor set is perfectly meager in the transitive sense. We answer a question of Nowik and Tsaban, whether of the same assertion holds for each Hurewicz set with no copy of the Cantor set inside. We solve this problem, under CH, in the negative.
This is a joint work with Tomasz Weiss and Lyubomyr Zdomskyy.
The research was funded by the National Science Centre, Poland and the Austrian Science Found under the Weave-UNISONO call in the Weave programme, project: Set-theoretic aspects of topological selections 2021/03/Y/ST1/00122
Location: A.4.1 C-19
Speaker: Piotr Szewczak (UKSW)
Title: Perfectly meager sets in the transitive sense and the Hurewicz property
Abstract: We work in the Cantor space with the usual group operation +. A set X is perfectly meager in the transitive sense if for any perfect set P there is an F-sigma set F containing X such that for every point t the intersection of t+F and P is meager in the relative topology of P. A set X is Hurewicz if for any sequence of increasing open covers of X one can select one set from each cover such that the chosen sets formulate a gamma-cover of X, i.e., an infinite cover such that each point from X belongs to all but finitely many sets from the cover. Nowik proved that each Hurewicz set which cannot be mapped continuously onto the Cantor set is perfectly meager in the transitive sense. We answer a question of Nowik and Tsaban, whether of the same assertion holds for each Hurewicz set with no copy of the Cantor set inside. We solve this problem, under CH, in the negative.
This is a joint work with Tomasz Weiss and Lyubomyr Zdomskyy.
The research was funded by the National Science Centre, Poland and the Austrian Science Found under the Weave-UNISONO call in the Weave programme, project: Set-theoretic aspects of topological selections 2021/03/Y/ST1/00122
Piotr Szewczak: Totally imperfect Menger sets
31/05/23 10:19
Tuesday, June 6, 2023 17:00
Location: room A.4.1 C-19
Speaker: Piotr Szewczak (UKSW)
Title: Totally imperfect Menger sets
Abstract: A set of reals \(X\) is Menger if for any countable sequence of open covers of \(X\) one can pick finitely many elements from every cover in the sequence such that the chosen sets cover \(X\). Any set of reals of cardinality smaller than the dominating number d is Menger and there is a non-Menger set of cardinality \(d\). By the result of Bartoszyński and Tsaban, in ZFC, there is a totally imperfect (with no copy of the Cantor set inside) Menger set of cardinality \(d\). We solve a problem, whether there is such a set of cardinality continuum. Using an iterated Sacks forcing and topological games we prove that it is consistent with ZFC that \(d \lt c\) and each totally imperfect Menger set has cardinality less or equal than \(d\).
This is a joint work with Valentin Haberl and Lyubomyr Zdomskyy.
The research was funded by the National Science Centre, Poland and the Austrian Science Found under the Weave-UNISONO call in the Weave programme, project: Set-theoretic aspects of topological selections 2021/03/Y/ST1/00122.
Location: room A.4.1 C-19
Speaker: Piotr Szewczak (UKSW)
Title: Totally imperfect Menger sets
Abstract: A set of reals \(X\) is Menger if for any countable sequence of open covers of \(X\) one can pick finitely many elements from every cover in the sequence such that the chosen sets cover \(X\). Any set of reals of cardinality smaller than the dominating number d is Menger and there is a non-Menger set of cardinality \(d\). By the result of Bartoszyński and Tsaban, in ZFC, there is a totally imperfect (with no copy of the Cantor set inside) Menger set of cardinality \(d\). We solve a problem, whether there is such a set of cardinality continuum. Using an iterated Sacks forcing and topological games we prove that it is consistent with ZFC that \(d \lt c\) and each totally imperfect Menger set has cardinality less or equal than \(d\).
This is a joint work with Valentin Haberl and Lyubomyr Zdomskyy.
The research was funded by the National Science Centre, Poland and the Austrian Science Found under the Weave-UNISONO call in the Weave programme, project: Set-theoretic aspects of topological selections 2021/03/Y/ST1/00122.
Piotr Szewczak: The Scheepers property and products of Menger spaces
10/03/17 08:01
Tuesday, March 14, 2017 17:15
Room: D1-215
Speaker: Piotr Szewczak
Title: The Scheepers property and products of Menger spaces
Abstract. A topological space \(X\) is Menger if for every sequence of open covers \(\mathcal{O}_1, \mathcal{O}_2,\ldots \) of the space \(X\), there are finite subfamilies \(\mathcal{F}_1\subseteq \mathcal{O}_1,\ \mathcal{F}_2\subseteq\mathcal{O}_2,\ldots \) such that their union is a cover of \(X\). If, in addition, for every finite subset \(F\) of \(X\) there is a natural number \(n\) with \(F\subseteq\bigcup\mathcal{F}_n\), then the space \(X\) is Scheepers. The above properties generalize \(\sigma\)-compactness, and Scheepers’ property is formally stronger than Menger’s property. It is consistent with ZFC that these properties are equal.
One of the open problems in the field of selection principles is to find the minimal hypothesis that the above properties can be separated in the class of sets of reals. Using purely
combinatorial approach, we provide examples under some set theoretic hypotheses. We apply obtained results to products of Menger spaces
This a joint work with Boaz Tsaban (Bar-Ilan University, Israel) and Lyubomyr Zdomskyy (Kurt Godel Research Center, Austria).
Room: D1-215
Speaker: Piotr Szewczak
Title: The Scheepers property and products of Menger spaces
Abstract. A topological space \(X\) is Menger if for every sequence of open covers \(\mathcal{O}_1, \mathcal{O}_2,\ldots \) of the space \(X\), there are finite subfamilies \(\mathcal{F}_1\subseteq \mathcal{O}_1,\ \mathcal{F}_2\subseteq\mathcal{O}_2,\ldots \) such that their union is a cover of \(X\). If, in addition, for every finite subset \(F\) of \(X\) there is a natural number \(n\) with \(F\subseteq\bigcup\mathcal{F}_n\), then the space \(X\) is Scheepers. The above properties generalize \(\sigma\)-compactness, and Scheepers’ property is formally stronger than Menger’s property. It is consistent with ZFC that these properties are equal.
One of the open problems in the field of selection principles is to find the minimal hypothesis that the above properties can be separated in the class of sets of reals. Using purely
combinatorial approach, we provide examples under some set theoretic hypotheses. We apply obtained results to products of Menger spaces
This a joint work with Boaz Tsaban (Bar-Ilan University, Israel) and Lyubomyr Zdomskyy (Kurt Godel Research Center, Austria).
Piotr Szewczak: Products of Menger spaces
18/11/15 18:41
Tuesday, November 24, 2015 17:15
Room: D1-215
Speaker: Piotr Szewczak (Cardinal Stefan Wyszyński University in Warsaw); Coauthor: Boaz Tsaban (Bar-Ilan University, Israel)
Title: Products of Menger spaces
Abstract. A topological space \(X\) is Menger if for every sequence of open covers \(O_1, O_2, \ldots\) there are finite subfamilies \(F_1\) of \(O_1\), \(F_2\) of \(O_2\), . . . such that their union is a cover of \(X\). The above property generalizes sigma-compactness.
One of the major open problems in the field of selection principles is whether there are, in ZFC, two Menger sets of real numbers whose product is not Menger. We provide examples under various set theoretic hypotheses, some being weak portions of the Continuum Hypothesis, and some violating it. The proof method is new.
Room: D1-215
Speaker: Piotr Szewczak (Cardinal Stefan Wyszyński University in Warsaw); Coauthor: Boaz Tsaban (Bar-Ilan University, Israel)
Title: Products of Menger spaces
Abstract. A topological space \(X\) is Menger if for every sequence of open covers \(O_1, O_2, \ldots\) there are finite subfamilies \(F_1\) of \(O_1\), \(F_2\) of \(O_2\), . . . such that their union is a cover of \(X\). The above property generalizes sigma-compactness.
One of the major open problems in the field of selection principles is whether there are, in ZFC, two Menger sets of real numbers whose product is not Menger. We provide examples under various set theoretic hypotheses, some being weak portions of the Continuum Hypothesis, and some violating it. The proof method is new.