We examine dimensional types of scattered P-spaces of weight ω 1. Such spaces can be embedded int... more 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 .
In this paper we introduce the notion of the center of distances of a metric space, which is requ... more 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.
Characterizations of skeletally Dugundji spaces and Dugundji spaces are given in terms of club co... more 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.
For any infinite subset X of the rationals and a subset F ⊆ X which has no isolated points in X w... more For any infinite subset X of the rationals and a subset F ⊆ X which has no isolated points in X we construct a function f: X → X such that f(f(x))=x for each x∈ X and F is the set of discontinuity points of f.
Following Frink's characterization of completely regular spaces, we say that a regular T_1-sp... more 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.
The paper fills gaps in knowledge about Kuratowski operations which are already in the literature... more 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 (V) which are generalizations of the ideal (v 0). We formulate couterparts... more We consider ideals d 0 (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 (V)) ≤ add(d 0 (V)) + .
We try to explain the differences between the concepts of stratifiable space and $\varkappa$-metr... more 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.
For any infinite subset X of the rationals and a subset F C X which has no isolated points in X w... more For any infinite subset X of the rationals and a subset F C X which has no isolated points in X we construct a function / : X -> X such that f{f(x)) = x for each x C X and F is the set of discontinuity points of/.
We prove that the Niemytzki plane is κ-metrizable and we try to explain the differences between t... more We prove that the Niemytzki plane is κ-metrizable and we try to explain the differences between the concepts of a stratifiable space and a κ-metrizable space. Also, we give a characterisation of κ-metrizable spaces which is modelled on the version described by Chigogidze.
Dimensional types of metric scattered spaces are investigated. Revised proofs of Mazurkiewicz-Sie... more Dimensional types of metric scattered spaces are investigated. Revised proofs of Mazurkiewicz-Sierpi\'nski and Knaster-Urbanik theorems are presented. Embeddable properties of countable metric spaces are generalized onto uncountable metric $\sigma$-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 $X \subseteq \omega_1$ is a stationary set, then the poset formed from dimensional types of subspaces of $X$ contains uncountable anti-chains and uncountable strictly descending chains.
The $\sigma$-ideal $(v^0)$ is associated with the Silver forcing, see \cite{bre}. Also, it consti... more The $\sigma$-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 $h=\omega_1$ implies that $(v^0)$ has the ideal type $(\frak c, \omega_1,\frak c)$.
It is showed that any compact space X is I-favorable if, and only if X can be representing as a l... more It is showed that any 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. [Compact openly generated spaces.] In several papers [13], [14] and [15], E.V. Shchepin considered a few classes of compact spaces. Among others, he introduced the class of compact openly generated spaces. A compact space X is called openly generated, whenever X is a limit of a σ-complete inverse system of metrizable spaces with open bonding maps. Since Ivanov's result [9]: A compact space X is openly generated if, and only if its superextension is a Dugundji space; Shchepin established that the classes of openly generated compact spaces and of κ-metrizable spaces are the same, see Theorem 21 in [15]. [Limits of inverse systems with skeletal bonding maps.] We consider the class of topological (compact) spaces which widen openly generated spaces. Our's class consists of limits of σ-complete ...
In this paper we establish compactness results of multiscale and very weak multiscale type for se... more In this paper we establish compactness results of multiscale and very weak multiscale type for sequences bounded in L 2 (0, T ; H 1 0 (Ω)), fulfilling a certain condition. We apply the results in the homogenization of the parabolic partial differential equation ε p ∂ t uε(x, t) − ∇ • (a(xε −1 , xε −2 , tε −q , tε −r)∇uε(x, t)) = f (x, t), where 0 < p < q < r. The homogenization result reveals two special phenomena, namely that the homogenized problem is elliptic and that the matching for which the local problem is parabolic is shifted by p, compared to the standard matching that gives rise to local parabolic problems.
We investigate the category of discrete topological spaces, with emphasis on inverse systems of h... more We investigate the category of discrete topological spaces, with emphasis on inverse systems of height ω1. Their inverse limits belong to the class of P -spaces, which allows us to explore dimensional types of these spaces.
We introduce the notion of a center of distances of a metric space and use it in a generalization... more 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 3 4 , 1 2 , 3 16 , 1 8 ,. .. , 3 4 n , 2 4 n ,. . ., and for other related subsets of the reals. Keywords Cantorval • Center of distances • von Neumann's theorem • Set of subsums • Digital representation Mathematics Subject Classification 40A05 • 11B05 • 28A75 Wojciech Bielas and Marta Walczyńska acknowledge support from GAČR project 16-34860L and RVO: 67985840.
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Papers by Szymon Plewik