Szymon Plewik

On the ideal (v 0)

The σ-ideal (v0) is associated with the Silver forcing, see [5]. Also, it constitutes the family of all completely doughnut null sets, see [9]. We introduce segment topologies to state some resemblances of (v0) to the family of Ramsey null sets. To describe add(v0) we adopt a proof of Base Matrix Lemma. Consistent results are stated, too. Halbeisen’s conjecture cov(v0) = add(v0) is confirmed under the hypothesis t = min{cf(c), r}. The hypothesis cov(v0) = ω1 implies that (v0) has the ideal type (c, ω1, c).

On the center of distances

We introduce the notion of a center of distances of a metric space and use it in a generalization of the theorem by John von Neumann on permutations of two sequences with the same set of cluster points in a compact metric space. This notion is also used to study sets of subsums of some sequences of positive reals, as well for some impossibility proofs. We compute the center of distances of the Cantorval, which is the set of subsums of the sequence

Game theoretic approach to skeletally Dugundji and Dugundji spaces

\frac{3}{4}, \frac{1}{2}, \frac{3}{16}, \frac{1}{8}, \ldots , \frac{3}{4^n}, \frac{2}{4^n}, \ldots

Ideals which generalize (v 0)

34,12,316,18,…,34n,24n,…, and for other related subsets of the reals.

Inverse systems and I-favorable spaces

We introduce and investigate the class of skeletally Dugundji spaces as a skeletal analogue of Dugundji space. Our main result states that the following conditions are equivalent for a given space X: (i) X is skeletally Dugundji; (ii) every compactification of X is co-absolute to a Dugundji space; (iii) every C*-embedding of the absolute p(X) in another space is strongly π-regular; (iv) X has a multiplicative lattice in the sense of Shchepin [Shchepin E.V., Topology of limit spaces with uncountable inverse spectra, Uspekhi Mat. Nauk, 1976, 31(5), 191–226 (in Russian)] consisting of skeletal maps.

SKELETAL MAPS AND I-FAVORABLE SPACES

Abstract Characterizations of skeletally Dugundji spaces and Dugundji spaces are given in terms of club collections, consisting of countable families of co-zero sets. For example, a Tychonoff space X is skeletally Dugundji if and only if there exists an additive c-club on X. Dugundji spaces are characterized by the existence of additive d-clubs.

Game approach to universally Kuratowski–Ulam spaces

The paper fills gaps in knowledge about Kuratowski operations which are already in the literature. The Cayley table for these operations has been drawn up. Techniques, using only paper and pencil, to point out all semigroups and its isomorphism types are applied. Some results apply only to topology, and one cannot bring them out, using only properties of the complement and a closure-like operation. The arguments are by systematic study of possibilities.

On regular but not completely regular spaces

Countable products of finite discrete spaces with more than one point and ideals generated by Marczewski-Burstin bases (assigned to trimmed trees) are examined, using machinery of base tree in the sense of B. Balcar and P. Simon. Applying Kulpa-Szymanski Theorem, we prove that the covering number equals to the additivity or the additivity plus for each of the ideals considered.