May 2020

# 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 Mikołaj 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 Mikołaj Krupski. The preprint containing these results can be found here: https://arxiv.org/abs/2002.07423

# Lyubomyr Zdomskyy: Menger and Hurewicz spaces: products and applications to forcing

22/05/20 06:37

Tuesday, May 26, 2020 17:15

*Location:***Zoom.us**: if you want to participate please contact organizers*Lyubomyr Zdomskyy (KGHR, Vienna)*

Speaker:Speaker:

*Title*: Menger and Hurewicz spaces: products and applications to forcing*Abstract*. This talk will be devoted to (products of) Menger and Hurewicz spaces and their connections to forcing and mad families. In particular, we shall show that in the Laver model, each mad family can be destroyed by a ccc poset preserving the ground model reals unbounded and splitting. It is an important open problem whether the same follows from CH.# Włodzimierz Charatonik: Degree of non local connectedness

14/05/20 18:23

Tuesday, May 19, 2020 18:15

*Location:***Zoom.us**: if you want to participate please contact organizers*Włodzimierz Charatonik*

Speaker:Speaker:

*Title*: Degree of non local connectedness*Abstract*. For a given continuum \(X\) we assign a cardinal number or a symbol \(\infty\) \(\tau(X)\) called degree of non local connectedness. The number \(\tau(X)\) cannot be increased by a continuous image; we show theorems about cartesian products, hyperspaces etc. Based on an article by Janusz J. Charatonik and Włodzimierz J. Charatonik# Włodzimierz Charatonik: Zero-dimensional compact metric spaces \(X\) whose squares \(X^2\) are homeomorphic to \(X\)

14/05/20 18:20

Tuesday, May 19, 2020 17:15

*Location:***Zoom.us**: if you want to participate please contact organizers*Włodzimierz Charatonik*

Speaker:Speaker:

*Title*: Zero-dimensional compact metric spaces \(X\) whose squares \(X^2\) are homeomorphic to \(X\)*Abstract*. We construct a family of cardinality \(\omega_1\) of (non homeomorphic) countable compact metric spaces \(X\) such that \(X\) is homeomorphic to \(X^2\). Based on an article by Włodzimierz J. Charatonik and Sahika Sahan.# Taras Banakh: Set Theoretic Problems in Large-Scale Topology

05/05/20 21:56

Tuesday, May 12, 2020 17:15

More details can be found in the preprints (written jointly with Igor Protasov):

https://arxiv.org/abs/2004.01979

https://arxiv.org/abs/2002.08800

*Location:***Zoom.us**: if you want to participate please contact organizers*Taras Banakh*

Speaker:Speaker:

*Title*: Set Theoretic Problems in Large-Scale Topology*Abstract*. We survey some set-theoretic problems appearing in large-scale topology.More details can be found in the preprints (written jointly with Igor Protasov):

https://arxiv.org/abs/2004.01979

https://arxiv.org/abs/2002.08800