Pierwszy wykład z cyklu Wykłady ID-UB

Abstrakt

Definition 1. A subset of reals \(X\subset \mathbb R\) is said to be a \(\Delta\)-set if \(X\) has the following property \(\Delta\): for every decreasing sequence \(\{D_n: n \in \omega\}\) of subsets of \(X\) with empty intersection, there is a decreasing sequence \(\{V_n: n \in \omega\}\) consisting of open subsets of \(X\), also with empty intersection, and such that \(D_n \subset V_n\) for every \(n \in \omega\).

The existence of an uncountable \(\Delta\)-set \(X\subset \mathbb{R}\) is one of the fundamental set-theoretical problems; it is independent of ZFC (due to results of F. Hausdorff and M.E. Rudin).

Definition 2. A locally convex space (lcs) \(E\) is called distinguished if the strong dual of \(E\) (i.e. the topological dual of \(E\) endowed with the strong topology) is barrelled (A. Grothendieck).

What two defined above notions from two completely different areas, one from Set Theory, another from Functional Analysis, can have in common?

By \(C_{p}(X)\) we mean the space of all real-valued continuous functions on a Tychonoff topological space \(X\) endowed with the topology of pointwise convergence.

Theorem 3 ([1]). Let \(X\) be a Tychonoff space. A lcs \(C_{p}(X)\) is distinguished if and only if for each \(f\in\mathbb{R}^{X}\) there is a bounded \(B\subset C_{p}(X)\) such that \(f\) belongs to the closure of \(B\) in \(\mathbb{R}^{X}\).

We say that a Tychonoff space \(X\) is a \(\Delta\)-space if \(X\) satisfies property \(\Delta\), as in Definition 1 above. Quite recently we have shown that the notion of \(\Delta\)-spaces plays a key role in the study of distinguished \(C_p\)-spaces [2].

Characterization Theorem 4 ([2]). Let \(X\) be a Tychonoff space. A lcs \(C_{p}(X)\) is distinguished if and only if \(X\) is a \(\Delta\)-space.

My talk will be devoted to the main results published in the joint with J. Kąkol papers [1,2,3]; and also my works [4,5].

References

[1] J. C. Ferrando, J. Kąkol, A. Leiderman, S. A. Saxon, Distinguished \(C_{p}(X)\) spaces, RACSAM 115:27 (2021).

[2] J. Kąkol, A. Leiderman, A characterization of \(X\) for which spaces \(C_p(X)\) are distinguished and its applications, Proc. Amer. Math. Soc., series B, 8 (2021), 86-99.

[3] J. Kąkol, A. Leiderman, Basic properties of \(X\) for which the space \(C_p(X)\) is distinguished, Proc. Amer. Math. Soc., series B, 8 (2021), 267-280.

[4] A. Leiderman, V.V. Tkachuk, Pseudocompact \(\Delta\)-spaces are often scattered, Monatsh. Math. 197 (2022), 493-503.

[5] A. Leiderman, P. Szeptycki, On \(\Delta\)-spaces, preprint.