[Settfa] [Set Theory]Tuesday 10 Sep, Jerzy Kakol - absence

[fabian]

Dear Wieslaw,

Unfortunately I will not come - I have an appointment at a doctor, 
specialist.
It is pitty because the topic seems very attractive to me.
Please, forward Jurek my best regards and excuse.

Marian



ne 2024-09-07 09:00, kubis at math.cas.cz napsal:
> Tuesday, 10 September 2024 - 10:00 to 11:30
> Place: IM, konírna
> 
>  Speaker: Jerzy Kakol, IM CAS and Adam Mickiewicz University, Poznań
> Title: A new characterization of compact scattered spaces X in terms
> of spaces C_p(X)
> 
>  Abstract
> For a Tychonoff space X by C_p(X) we denote the space of continuous
> real valued function on X endowed with the pointwise topology, and
> C(X) denotes the Banach space endowed with the uniform topology
> provided X is compact.
> 
> The classical two results characterizing compact scattered spaces in
> terms of C(X) and C_p(X) assert that a compact space X is scattered if
> and only if C(X) is an Asplund space (Namioka-Phelps) if and only if
> C_p(X) is a Frechet-Urysohn space (Gerlits, Pytkkev). We provide
> another result of this type by showing the following Theorem: An
> infinite compact space X is scattered if and only if C_p(X) contains
> no closed \sigma-compact infinite-dimensional vector subspace if and
> only if C_p(X) contains no infinite-dimensional vector subspace
> admitting a fundamental sequence of bounded sets if and only if every
> topological vector subspace of C_p(X) is bornological. Above  Theorem
> fails if X is  scattered but not compact. These results are also
> motivated by a remarkable theorem of Velichko's stating that for an
> infinite Tychonoff space X the space C_p(X) is not \sigma-compact.
> Several illustrating examples involving spaces c_0, \ell_{\infty} and
> the space Lip_0(M) with the pointwise topology are provided and
> discussed.
> 
>  For more information see the seminar web page at
> https://www.math.cas.cz/index.php/events/seminar/6
> 
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