arXiv subjects
Publications and source records attributed to Szymon Plewik.
We first present a reduction of properties of compact sets of real numbers to properties of countable orders. Then discuss a variant of homogeneity of compact subsets of real numbers, focusing on the family of $t$-sets. Finally, we prove that there are exactly $\omega_1$ many non-homeomorphic $t$-sets.
Motivated by results of J. R. Kline and R. L. Moore (1919) that a compact subset of the plane, homeomorphic to a subset of the reals, lies on the arc, we give a purely topological characterisation of compact sets of the reals. This allows us to reduce investigations of Cantorvals to properties of countable linear orders and to show, applying the Mazurkiewicz--Sierpi\'nski Theorem (1920), that there exist continuum many non-homeomorphic L-Cantorvals.
We examine dimensional types of scattered $P$-spaces of weight $ω_1$. Such spaces can be embedded into $ω_2$. There are established similarities between dimensional types of scattered separable metric spaces and dimensional types of $P$-spaces of weight $ω_1$ with Cantor--Bendixson rank less than $ω_1$.
We investigate the category of discrete topological spaces, with emphasis on inverse systems of height $ω_1$. Their inverse limits belong to the class of $P$-spaces, which allows us to explore dimensional types of these spaces.
We try to explain the differences between the concepts of stratifiable space and $\varkappa$-metrizable space. In particular, we give a characterization of $\varkappa$-metrizable spaces which is modelled on Chigogidze's characterization. Moreover, we present a $\varkappa$-metric for the Niemytzki plane, using the properties of the Euclidean metric.
Following Frink's characterization of completely regular spaces, we say that a regular T_1-space is an RC-space whenever the family of all regular open sets constitutes a regular normal base. Normal spaces are RC-spaces and there exist completely regular spaces which are not RC-spaces. So the question arises, which of the known examples of completely regular and not normal spaces are RC-spaces. We show that the Niemytzki plane and the Sorgenfrey plane are RC-spaces.
We present how to obtain non-comparable regular but not completely regular spaces. We analyze a generalization of Mysior's example, extracting its underlying purely set-theoretic framework. This enables us to build simple counterexamples, using the Niemytzki plane, the Songefrey plane or Lusin gaps.
In this paper we introduce the notion of the center of distances of a metric space, which is required for a generalization of the theorem by J. von Neumann about permutations of two sequences with the same set of cluster points in a compact metric space. Also, the introduced notion is 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 $\frac34, \frac12, \frac3{16}, \frac18, \ldots , \frac3{4^n}, \frac2{4^n}, \ldots$, and also for some related subsets of the reals.
Dimensional types of metric scattered spaces are investigated. Revised proofs of Mazurkiewicz-Sierpiński and Knaster-Urbanik theorems are presented. Embeddable properties of countable metric spaces are generalized onto uncountable metric $σ$-discrete spaces. Some related topics are also explored. For example: For each infinite cardinal number $\frak m$, there exist $2^{\frak m}$ many non-homeomorphic metric scattered spaces of the cardinality $\frak m $; If $X \subseteq ω_1$ is a stationary set, then the poset formed from dimensional types of subspaces of $X$ contains uncountable anti-chains and uncountable strictly descending chains.
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.
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 isomorphic types are applied. Some results apply only to topology, one can not bring them out, using only properties of the complement and a closure-like operation. The arguments are by systematic study of possibilities.
We consider ideals $d^0(\mathcal{V})$ which are generalizations of the ideal $(v^0)$. We formulate couterparts of Hadamard's theorem. Then, adopting the base tree theorem and applying Kulpa-Szymański Theorem, we obtain $ cov(d^0(\mathcal{V}))\leq add(d^0(\mathcal{V}))^+$.
We consider a question: Can a given AD-family be ADR for two orthogonal uncountable towers? If $b > ω_1$, then we rebuilt any AD-family of the cardinality $ω_1$ onto a Hausdorff pre-gap. Moreover, if a such AD-family is a Luzin gap, then we obtain a Hausdorff gap. Under $b = ω_1$, a similar rebuilding is impossible.
A compact space X is I-favorable if, and only if X can be representing as a limit of $σ$-complete inverse system of compact metrizable spaces with skeletal bonding maps.
The $σ$-ideal $(v^0)$ is associated with the Silver forcing, see \cite{bre}. Also, it constitutes the family of all completely doughnut null sets, see \cite{hal}. We introduce segments and $*$-segments topologies, to state some resemblances of $(v^0)$ to the family of Ramsey null sets. To describe $add(v^0)$ we adopt a proof of Base Matrix Lemma. Consistent results are stated, too. Halbeisen's conjecture $cov(v^0) = add(v^0)$ is confirmed under the hypothesis $t= \min \{\cf (\frak c), r\} $. The hypothesis $h=ω_1$ implies that $(v^0)$ has the ideal type $(\frak c, ω_1,\frak c)$.
For any infinite subset $X$ of the rationals and a subset $F \subseteq X$ which has no isolated points in $X$ we construct a function $f: X \to X$ such that $f(f(x))=x$ for each $x\in X$ and $F $ is the set of discontinuity points of $f$.
C-cross topologies are introduced. Modifcations of the Kuratowski-Ulam Theorem are considered. Cardinal invariants add, cof, cov and non with respect to meager or nowhere dense subsets are compared. Remarks on invariants cof(nwdY) are mentioned for dense subspaces Y of X.