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Tuesday, 10 September 2024 - 10:00 to 11:30 <br />
Place: IM, konírna
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Speaker: Jerzy Kakol, IM CAS and Adam Mickiewicz University, Poznań<br />
Title: A new characterization of compact scattered spaces X in terms of spaces C_p(X)
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Abstract <br />
<p>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.</p><p>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.</p><p><br></p>
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For more information see the seminar web page at <br />
https://www.math.cas.cz/index.php/events/seminar/6
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Set Theory and Analysis mailing list <br />
settfa@math.cas.cz <br />
https://list.math.cas.cz/listinfo/settfa@math.cas.cz
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