Witold Marciszewski

# Witold Marciszewski: On \(\omega\)-Corson compact spaces and related classes of Eberlein compacta

02/11/23 10:39

Friday, November 3, 2023 16:15

A compact space \(K\) is \(\omega\)-Corson compact if, for some set \(\Gamma\), \(K\) is homeomorphic to a subset of the \(\sigma\)-product of real lines \(\sigma(\mathbb{R}^\Gamma)\), i.e. the subspace of the product \(\mathbb{R}^\Gamma\) consisting of functions with finite supports. Clearly, every \(\omega\)-Corson compact space is Eberlein compact.

We will present a characterization of \(\omega\)-Corson compact spaces, and some other results concerning this class of spaces and related classes of Eberlein compacta.

This is a joint research with Grzegorz Plebanek and Krzysztof Zakrzewski, see

https://arxiv.org/abs/2107.02513

*Location:*room 601, Mathematical Institute, University of Wroclaw*Witold Marciszewski (MIM UW)*

Speaker:Speaker:

*Title*: On \(\omega\)-Corson compact spaces and related classes of Eberlein compacta*Abstract*: Recall that a compact space \(K\) is Eberlein compact if it can be embedded into some Banach space X equipped with the weak topology; equivalently, for some set \(\Gamma\), \(K\) can be embedded into the space \(c_0( \Gamma)\), endowed with the pointwise convergence topology.A compact space \(K\) is \(\omega\)-Corson compact if, for some set \(\Gamma\), \(K\) is homeomorphic to a subset of the \(\sigma\)-product of real lines \(\sigma(\mathbb{R}^\Gamma)\), i.e. the subspace of the product \(\mathbb{R}^\Gamma\) consisting of functions with finite supports. Clearly, every \(\omega\)-Corson compact space is Eberlein compact.

We will present a characterization of \(\omega\)-Corson compact spaces, and some other results concerning this class of spaces and related classes of Eberlein compacta.

This is a joint research with Grzegorz Plebanek and Krzysztof Zakrzewski, see

https://arxiv.org/abs/2107.02513

# Witold Marciszewski: On zero-dimensional subspaces of Eberlein compacta

12/04/21 17:02

Tuesday, April 13, 2021 17:00

The talk is based on the paper "On two problems concerning Eberlein compacta": http://arxiv.org/abs/2103.03153

*Location:***Zoom.us**: if you want to participate please contact organizers*Witold Marciszewski (University of Warsaw)*

Speaker:Speaker:

*Title*: On zero-dimensional subspaces of Eberlein compacta*Abstract*: Let us recall that a compact space K is Eberlein compact if it can be embedded into some Banach space X equipped with the weak topology. Our talk will be devoted to the known problem of the existence of nonmetrizable compact spaces without nonmetrizable zero-dimensional closed subspaces. Several such spaces were obtained using some additional set-theoretic assumptions. Recently, P. Koszmider constructed the first such example in ZFC. We investigate this problem for the class of Eberlein compact spaces. We construct such Eberlein compacta, assuming the existence of a Luzin set. We also show that it is consistent with ZFC that each Eberlein compact space of weight greater than \(\omega_1\) contains a nonmetrizable closed zero-dimensional subspace.The talk is based on the paper "On two problems concerning Eberlein compacta": http://arxiv.org/abs/2103.03153

# Witold Marciszewski: On countable dense homogeneous topological vector spaces

27/05/20 11:18

Tuesday, June 2, 2020 17:15

This is a joint work with Tadek Dobrowolski and MikoĊaj Krupski. The preprint containing these results can be found here: https://arxiv.org/abs/2002.07423

*Location:***Zoom.us**: if you want to participate please contact organizers*Witold Marciszewski (Uniwersytet Warszawski)*

Speaker:Speaker:

*Title*: On countable dense homogeneous topological vector spaces*Abstract*. Recall that a topological space X is countable dense homogeneous (CDH) if X is separable, and given countable dense subsets D,E of X, there is an autohomeomorphism of X mapping D onto E. This is a classical notion tracing back to works of Cantor, Frechet and Brouwer. The canonical examples of CDH spaces include the Cantor set, the Hilbert cube, and all separable Banach spaces. All Borel, but not closed linear subspaces of Banach spaces are not CDH. By \(C_p(X)\) we denote the space of all continuous real-valued functions on a Tikhonov space X, endowed with the pointwise topology. V. Tkachuk asked if there exists a nondiscrete space X such that \(C_p(X)\) is CDH. Last year R. Hernandez Gutierrez gave the first consistent example of such a space X. He has asked whether a metrizable space X must be discrete, provided \(C_p(X)\) is CDH. We answer this question in the affirmative. Actually, combining our theorem with earlier results, we prove that, for a metrizable space X, \(C_p(X)\) is CDH if and only if X is discrete of cardinality less than pseudointersection number \(\mathfrak p\). We also prove that every CDH topological vector space X is a Baire space. This implies that, for an infinite-dimensional Banach space E, both spaces (E,w) and (E*,w*) are not CDH. We generalize some results of Hrusak, Zamora Aviles, and Hernandez Gutierrez concerning countable dense homogeneous products.This is a joint work with Tadek Dobrowolski and MikoĊaj Krupski. The preprint containing these results can be found here: https://arxiv.org/abs/2002.07423