Double Session at Fields Institute (Antonio Aviles and Istvan Juhasz)
Place: Fields Institute (Stewart Library)
Date and time: 28-11-2014 from 12:30 to 15:00
Speaker 1: Antonio Aviles (12:30-13:30)
Title: A combinatorial lemma about cardinals $\aleph_n$ and its applications on Banach spaces
Abstract: The lemma mentioned in the title was used by Enflo and Rosenthal to show that the Banach space $L_p[0,1]^\Gamma$ does not have an unconditional basis when $|\Gamma|\geq \aleph_\omega$. In a joint work with Witold Marciszewski, we used some variation of it to show that there are no extension operators between balls of different radii in nonseparable Hilbert spaces.
Speaker 2: Istvan Juhasz (13:40 to 15:00)
Title: Lindelof spaces of small extent are $\omega$-resolvable
Abstract: I intend to present the proof of the following result, joint with L. Soukup and Z. Szentmiklossy: Every regular space $X$ that satisfies $\Delta(X) > e(X)$ is $\omega$-resolvable, i.e. contains infinitely many pairwise disjoint dense subsets. Here $\Delta(X)$, the dispersion character of $X$, is the smallest size of a nonempty open set in $X$ and $e(X)$, the extent of $X$, is the supremum of the sizes of all closed-and-discrete subsets of $X$. In particular, regular Lindelof spaces of uncountable dispersion character are $\omega$-resolvable.
This improves some results of Pavlov and of Filatova, respectively, concerning Malychin’s problem if regular Lindelof spaces of uncountable dispersion character are resolvable at all. The question if regular Lindelof spaces of uncountable dispersion character are maximally resolvable, i.e. $\Delta(X)$-resolvable, remains wide open.
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