October 2015
Julia Wódka: Comparison of some families of real functions in sense of porosity
27/10/15 17:13
Tuesday, November 10, 2015 17:15
Room: D1-215
Speaker: Julia Wódka
Title: Comparison of some families of real functions in sense of porosity
Abstract. We consider set \(\mathbb{R}^\mathbb{R}\) with uniform convergence metric, i.e:
\[\rho(f,g)=\min\{1,\sup\limits_{x\in\mathbb{R}}|f(x)-g(x)|\}\quad \text{for \(f,g\in\mathbb{R}^\mathbb{R}\)}\]
and the following subsets of \(\mathbb{R}^\mathbb{R}\):
The aim of this is to compare this sets in terms of porosity.
Let \((X,d)\) be a metric space, \(x\in A\subset M\), and \(r\in\mathbb{R}_+\). We define
\[\gamma(x,r,M)=\sup\{{t\geq 0}:\ \exists_{z\in M} B(z,t)\subset B(x,r)\setminus M\}\]
and
\[p^u(M, x)=2\limsup\limits_{t\to r^+}\frac{\gamma(x,r,M)}{r}.\]
\[p_l(M, x)=2\liminf\limits_{t\to r^+}\frac{\gamma(x,r,M)}{r}.\]
Quantity \(p^u(M,x)\) is called upper porosity of \(M\) at the point \(x\). We say that \(M\) is upper \(p-\)porous if \(p=\inf\{p^u(M,x):\ x\in M\}>0\).
Analogously we define lower porosity.
Room: D1-215
Speaker: Julia Wódka
Title: Comparison of some families of real functions in sense of porosity
Abstract. We consider set \(\mathbb{R}^\mathbb{R}\) with uniform convergence metric, i.e:
\[\rho(f,g)=\min\{1,\sup\limits_{x\in\mathbb{R}}|f(x)-g(x)|\}\quad \text{for \(f,g\in\mathbb{R}^\mathbb{R}\)}\]
and the following subsets of \(\mathbb{R}^\mathbb{R}\):
- Darboux functions (\(f\in\mathscr{D}\) if whenever \(a < b\) and \(y\) is a number between \(f(a)\) and \(f(b)\), there exists an \(x_0\in(a, b)\) such that \(f(x_0) = y\)).
- quasi-continuous functions (\(f\in\mathscr{Q}\) if it is quasi-continuous at any point \(x\in\mathbb{R}\)).
Function \(f\) is quasi-continuous at \(x \in \mathbb{R}\) if for any open interval \(I\ni x\) and each \(\varepsilon>0\) there exists a nontrivial interval \(J\subset I\) such that \({\rm diam} (f[J\cup \{x\}]) <\varepsilon\). - Świątkowski functions ( \(f \in \mathscr{\acute S}\) if for all \( a < b \) with \(f(a) \ne f(b)\), there is a \(y\) between \(f(a)\) and \(f(b)\) and an \(x\in(a,b) \cap \mathcal{C}(f)\) such that \(f(x)=y\), where \(\mathcal{C}(f)\) denotes the set of all continuity points of function \(f\)).
- Świątkowski functions (\(f\in\mathscr{\acute S}_s\) if for all \(a < b\) and each \(y\) between \(f(a)\) and \(f(b)\) there is an \(x\in(a,b) \cap \mathcal{C}(f)\) such that \(f(x)=y\)).
The aim of this is to compare this sets in terms of porosity.
Let \((X,d)\) be a metric space, \(x\in A\subset M\), and \(r\in\mathbb{R}_+\). We define
\[\gamma(x,r,M)=\sup\{{t\geq 0}:\ \exists_{z\in M} B(z,t)\subset B(x,r)\setminus M\}\]
and
\[p^u(M, x)=2\limsup\limits_{t\to r^+}\frac{\gamma(x,r,M)}{r}.\]
\[p_l(M, x)=2\liminf\limits_{t\to r^+}\frac{\gamma(x,r,M)}{r}.\]
Quantity \(p^u(M,x)\) is called upper porosity of \(M\) at the point \(x\). We say that \(M\) is upper \(p-\)porous if \(p=\inf\{p^u(M,x):\ x\in M\}>0\).
Analogously we define lower porosity.
Aleksander Cieślak: On nonmeasurable subsets of \(\mathbb{R}\) and \(\mathbb{R}^2\)
21/10/15 21:14
Tuesday, October 27, 2015 17:15
Room: D1-215
Speaker: Aleksander Cieślak
Title: On nonmeasurable subsets of \(\mathbb{R}\) and \(\mathbb{R}^2\)
Abstract. I would like to present some results connected with the existence of a subset \(X\) of the square \([0,1]^2\) with the property that for any line \(L\) outside \([0,1]^2\) the projection \(\pi_L[X]\) is completely nonmeasurable in some interval with respect to selected \(\sigma\)-ideal with Borel base on the line \(L\).
Moreover, I will discuss the existence of large midpoint-free subsets of arbitrary subset of the real line.
Room: D1-215
Speaker: Aleksander Cieślak
Title: On nonmeasurable subsets of \(\mathbb{R}\) and \(\mathbb{R}^2\)
Abstract. I would like to present some results connected with the existence of a subset \(X\) of the square \([0,1]^2\) with the property that for any line \(L\) outside \([0,1]^2\) the projection \(\pi_L[X]\) is completely nonmeasurable in some interval with respect to selected \(\sigma\)-ideal with Borel base on the line \(L\).
Moreover, I will discuss the existence of large midpoint-free subsets of arbitrary subset of the real line.
Antonio Aviles: Boolean algebras obtained by push-out iteration
12/10/15 15:58
Tuesday, October 20, 2015 17:15
Room: D1-215
Speaker: Antonio Aviles (University of Murcia)
Title: Boolean algebras obtained by push-out iteration
Abstract. We discuss the notion of push-out in the category of Boolean algebras, and we describe a method of constructing Boolean algebras by transfinite iterative push-outs. Under CH and in a model obtained by adding \(\aleph_2\) Cohen reals to a model of CH, \(P(\omega)/fin\) is such an algebra.
Room: D1-215
Speaker: Antonio Aviles (University of Murcia)
Title: Boolean algebras obtained by push-out iteration
Abstract. We discuss the notion of push-out in the category of Boolean algebras, and we describe a method of constructing Boolean algebras by transfinite iterative push-outs. Under CH and in a model obtained by adding \(\aleph_2\) Cohen reals to a model of CH, \(P(\omega)/fin\) is such an algebra.
Damian Sobota: The Nikodym property and cardinal invariants of the continuum
12/10/15 11:21
Tuesday, October 13, 2015 17:15
Room: D1-215
Speaker: Damian Sobota
Title: The Nikodym property and cardinal invariants of the continuum
Abstract. A Boolean algebra \(\mathcal{A}\) is said to have the Nikodym property if every sequence \((\mu_n)\) of measures on \(\mathcal{A}\) which is elementwise bounded (i.e. \(\sup_n|\mu_n(a)|<\infty\) for every \(a\in\mathcal{A}\)) is uniformly bounded (i.e. \(\sup_n\|\mu_n\|<\infty\)). The property is closely related to the classical Banach-Steinhaus theorem for Banach spaces.
My recent study concerns the problem how (and whether at all) we can describe the structure of the class of Boolean algebras with the Nikodym property in terms of well-known objects occuring inside \(\wp(\omega)\) or \(\omega^\omega\), e.g. countable Boolean algebras, dominating families, Lebesgue null sets etc. During my talk I will present an attempt to obtain such a description via families of antichains in countable subalgebras of \(\wp(\omega)\) having some special measure-theoretic properties.
Room: D1-215
Speaker: Damian Sobota
Title: The Nikodym property and cardinal invariants of the continuum
Abstract. A Boolean algebra \(\mathcal{A}\) is said to have the Nikodym property if every sequence \((\mu_n)\) of measures on \(\mathcal{A}\) which is elementwise bounded (i.e. \(\sup_n|\mu_n(a)|<\infty\) for every \(a\in\mathcal{A}\)) is uniformly bounded (i.e. \(\sup_n\|\mu_n\|<\infty\)). The property is closely related to the classical Banach-Steinhaus theorem for Banach spaces.
My recent study concerns the problem how (and whether at all) we can describe the structure of the class of Boolean algebras with the Nikodym property in terms of well-known objects occuring inside \(\wp(\omega)\) or \(\omega^\omega\), e.g. countable Boolean algebras, dominating families, Lebesgue null sets etc. During my talk I will present an attempt to obtain such a description via families of antichains in countable subalgebras of \(\wp(\omega)\) having some special measure-theoretic properties.