Tomasz Żuchowski
Tomasz Żuchowski: The Nikodym property and filters on \(\omega\). Part II
22/04/24 08:20
Tuesday, April 23, 2024 17:15
Location: A.4.1 C-19
Speaker: Tomasz Żuchowski
Title: The Nikodym property and filters on \(\omega\). Part II
Abstract: For a free filter \(F\) on \(\omega\), we consider the space \(N_F=\omega\cup\{p_F\}\), where every element of \(\omega\) is isolated and open neighborhoods of \(p_F\) are of the form \(A\cup\{p_F\}\) for \(A\in F\).
In this talk we will study the family \(\mathcal{AN}\) of such ideals \(\mathcal{I}\) on \(\omega\) that the space \(N_{\mathcal{I}^*}\) carries a sequence \(\langle\mu_n\colon n\in\omega\rangle\) of finitely supported signed measures satisfying \(\|\mu_n\|\rightarrow\infty\) and \(\mu_n(A)\rightarrow 0\) for every \(A\in Clopen(N_{\mathcal{I}^*})\). If \(\mathcal{I}\in\mathcal{AN}\) and \(N_{\mathcal{I}^*}\) is embeddable into the Stone space \(St(\mathcal{A})\) of a given Boolean algebra \(\mathcal{A}\), then \(\mathcal{A}\) does not have the Nikodym property.
Location: A.4.1 C-19
Speaker: Tomasz Żuchowski
Title: The Nikodym property and filters on \(\omega\). Part II
Abstract: For a free filter \(F\) on \(\omega\), we consider the space \(N_F=\omega\cup\{p_F\}\), where every element of \(\omega\) is isolated and open neighborhoods of \(p_F\) are of the form \(A\cup\{p_F\}\) for \(A\in F\).
In this talk we will study the family \(\mathcal{AN}\) of such ideals \(\mathcal{I}\) on \(\omega\) that the space \(N_{\mathcal{I}^*}\) carries a sequence \(\langle\mu_n\colon n\in\omega\rangle\) of finitely supported signed measures satisfying \(\|\mu_n\|\rightarrow\infty\) and \(\mu_n(A)\rightarrow 0\) for every \(A\in Clopen(N_{\mathcal{I}^*})\). If \(\mathcal{I}\in\mathcal{AN}\) and \(N_{\mathcal{I}^*}\) is embeddable into the Stone space \(St(\mathcal{A})\) of a given Boolean algebra \(\mathcal{A}\), then \(\mathcal{A}\) does not have the Nikodym property.
Tomasz Żuchowski: The Nikodym property and filters on \(\omega\). Part I
25/03/24 11:14
Tuesday, March 26, 2024 17:15
Location: A.4.1 C-19
Speaker: Tomasz Żuchowski
Title: The Nikodym property and filters on \(\omega\). Part I
Abstract: For a free filter \(F\) on \(\omega\), we consider the space \(N_F=\omega\cup\{p_F\}\), where every element of \(\omega\) is isolated and open neighborhoods of \(p_F\) are of the form \(A\cup\{p_F\}\) for \(A\in F\).
In this talk we will study the family \(\mathcal{AN}\) of such ideals \(\mathcal{I}\) on \(\omega\) that the space \(N_{\mathcal{I}^*}\) carries a sequence \(\langle\mu_n\colon n\in\omega\rangle\) of finitely supported signed measures satisfying \(\|\mu_n\|\rightarrow\infty\) and \(\mu_n(A)\rightarrow 0\) for every \(A\in Clopen(N_{\mathcal{I}^*})\). If \(\mathcal{I}\in\mathcal{AN}\) and \(N_{\mathcal{I}^*}\) is embeddable into the Stone space \(St(\mathcal{A})\) of a given Boolean algebra \(\mathcal{A}\), then \(\mathcal{A}\) does not have the Nikodym property.
Location: A.4.1 C-19
Speaker: Tomasz Żuchowski
Title: The Nikodym property and filters on \(\omega\). Part I
Abstract: For a free filter \(F\) on \(\omega\), we consider the space \(N_F=\omega\cup\{p_F\}\), where every element of \(\omega\) is isolated and open neighborhoods of \(p_F\) are of the form \(A\cup\{p_F\}\) for \(A\in F\).
In this talk we will study the family \(\mathcal{AN}\) of such ideals \(\mathcal{I}\) on \(\omega\) that the space \(N_{\mathcal{I}^*}\) carries a sequence \(\langle\mu_n\colon n\in\omega\rangle\) of finitely supported signed measures satisfying \(\|\mu_n\|\rightarrow\infty\) and \(\mu_n(A)\rightarrow 0\) for every \(A\in Clopen(N_{\mathcal{I}^*})\). If \(\mathcal{I}\in\mathcal{AN}\) and \(N_{\mathcal{I}^*}\) is embeddable into the Stone space \(St(\mathcal{A})\) of a given Boolean algebra \(\mathcal{A}\), then \(\mathcal{A}\) does not have the Nikodym property.
Tomasz Żuchowski: Nonseparable growth of omega supporting a strictly positive measure
14/03/16 18:57
Tuesday, March 15, 2016 17:15
Room: D1-215
Speaker: Tomasz Żuchowski
Title: Nonseparable growth of omega supporting a strictly positive measure
Abstract. We will construct in ZFC a compactification \(\gamma\omega\) of \(\omega\) such that its remainder \(\gamma\omega\backslash\omega\) is not separable and carries a strictly positive measure, i.e. measure positive on nonempty open subsets. Moreover, the measure on our space is defined by the asymptotic density of subsets of \(\omega\).
Our remainder is a Stone space of a Boolean subalgebra of Lebesgue measurable subsets of \(2^{\omega}\) containing all clopen sets.
Room: D1-215
Speaker: Tomasz Żuchowski
Title: Nonseparable growth of omega supporting a strictly positive measure
Abstract. We will construct in ZFC a compactification \(\gamma\omega\) of \(\omega\) such that its remainder \(\gamma\omega\backslash\omega\) is not separable and carries a strictly positive measure, i.e. measure positive on nonempty open subsets. Moreover, the measure on our space is defined by the asymptotic density of subsets of \(\omega\).
Our remainder is a Stone space of a Boolean subalgebra of Lebesgue measurable subsets of \(2^{\omega}\) containing all clopen sets.
Tomasz Żuchowski: Tukey types of orthogonal ideals
08/05/15 08:24
Tuesday, May 12, 2015 17:15
Room: D1-215
Speaker: Tomasz Żuchowski
Title: Tukey types of orthogonal ideals
Abstract. A partial order \(P\) is Tukey reducible to partial order \(Q\) when there exists a function \(f:P\to Q\) such that if \(A\) is a bounded subset of \(Q\) then \(f^{-1}[A]\) is a bounded subset of \(P\). The existence of such reduction is related to some cardinal invariants of considered orders. We will show Tukey reductions between some special ideals of subsets of \(\mathbb{N}\) with the inclusion order and other partial orders.
Room: D1-215
Speaker: Tomasz Żuchowski
Title: Tukey types of orthogonal ideals
Abstract. A partial order \(P\) is Tukey reducible to partial order \(Q\) when there exists a function \(f:P\to Q\) such that if \(A\) is a bounded subset of \(Q\) then \(f^{-1}[A]\) is a bounded subset of \(P\). The existence of such reduction is related to some cardinal invariants of considered orders. We will show Tukey reductions between some special ideals of subsets of \(\mathbb{N}\) with the inclusion order and other partial orders.