The dimension of metrizable subspaces of Eberlein compacta and Eberlein compactifications of metrizable spaces

Michael G. Charalambous

Displaying similar documents to “The dimension of metrizable subspaces of Eberlein compacta and Eberlein compactifications of metrizable spaces”

Module-valued functors preserving the covering dimension

Jan Spěvák (2015)

Commentationes Mathematicae Universitatis Carolinae

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We prove a general theorem about preservation of the covering dimension $dim$ by certain covariant functors that implies, among others, the following concrete results. If $G$ G is a pathwise connected...

Strong Cohomological Dimension

Jerzy Dydak, Akira Koyama (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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We characterize strong cohomological dimension of separable metric spaces in terms of extension of mappings. Using this characterization, we discuss the relation between strong cohomological dimension and (ordinal) cohomological dimension and give examples to clarify their gaps. We also show that $I n d_{G} X = d i m_{G} X$ if X is a separable metric ANR and G is a countable Abelian group. Hence $d i m_{ℤ} X = d i m X$ for any separable metric ANR X.

A study of universal elements in classes of bases of topological spaces

Dimitris N. Georgiou, Athanasios C. Megaritis, Inderasan Naidoo, Fotini Sereti (2021)

Commentationes Mathematicae Universitatis Carolinae

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The universality problem focuses on finding universal spaces in classes of topological spaces. Moreover, in “Universal spaces and mappings” by S. D. Iliadis (2005), an important method of constructing such universal elements in classes of spaces is introduced and explained in details. Simultaneously, in “A topological dimension greater than or equal to the classical covering dimension” by D. N. Georgiou, A. C. Megaritis and F. Sereti (2017), new topological dimension is introduced and...

Levi's forms of higher codimensional submanifolds

Andrea D&#039;Agnolo, Giuseppe Zampieri (1991)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Let $X ≅ C^{n}$ , let $M$ be a $C^{2}$ hypersurface of $X$ , $S$ be a $C^{2}$ submanifold of $M$ . Denote by $L_{M}$ the Levi form of $M$ at $z_{0} \in S$ . In a previous paper [3] two numbers $s^{\pm} (S, p)$ , $p \in {({\dot{T}}_{S}^{*} X)}_{z_{0}}$ are defined; for $S = M$ they are the numbers of positive and negative eigenvalues for $L_{M}$ . For $S \subset M$ , $p \in S \times_{M} \dot{T}^{*}_{S} X)$ , we show here that $s^{\pm} (S, p)$ are still the numbers of positive and negative eigenvalues for $L_{M}$ when restricted to $T_{z_{0}}^{C} S$ . Applications to the concentration in degree for microfunctions at the boundary are given.

Remainders of metrizable spaces and a generalization of Lindelöf Σ-spaces

A. V. Arhangel'skii (2011)

Fundamenta Mathematicae

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We establish some new properties of remainders of metrizable spaces. In particular, we show that if the weight of a metrizable space X does not exceed $2^{ω}$ , then any remainder of X in a Hausdorff compactification is a Lindelöf Σ-space. An example of a metrizable space whose remainder in some compactification is not a Lindelöf Σ-space is given. A new class of topological spaces naturally extending the class of Lindelöf Σ-spaces is introduced and studied. This leads to the following theorem:...

Homogeneity and non-coincidence of Hausdorff and box dimensions for subsets of ℝⁿ

Anders Nilsson, Peter Wingren (2007)

Studia Mathematica

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A class of subsets of ℝⁿ is constructed that have certain homogeneity and non-coincidence properties with respect to Hausdorff and box dimensions. For each triple (r,s,t) of numbers in the interval (0,n] with r < s < t, a compact set K is constructed so that for any non-empty subset U relatively open in K, we have $(d i m_{H} (U), {\underset{̲}{d i m}}_{B} (U), {\bar{d i m}}_{B} (U)) = (r, s, t)$ . Moreover, $2^{- n} \leq H^{r} (K) \leq 2 n^{r / 2}$ .

Maps with dimensionally restricted fibers

Vesko Valov (2011)

Colloquium Mathematicae

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We prove that if f: X → Y is a closed surjective map between metric spaces such that every fiber $f^{- 1} (y)$ belongs to a class S of spaces, then there exists an $F_{σ}$ -set A ⊂ X such that A ∈ S and $d i m f^{- 1} (y) ∖ A = 0$ for all y ∈ Y. Here, S can be one of the following classes: (i) M: e-dim M ≤ K for some CW-complex K; (ii) C-spaces; (iii) weakly infinite-dimensional spaces. We also establish that if S = M: dim M ≤ n, then dim f ∆ g ≤ 0 for almost all $g \in C (X,^{n + 1})$ .

Szpilrajn type theorem for concentration dimension

Jozef Myjak, Tomasz Szarek (2002)

Fundamenta Mathematicae

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Let X be a locally compact, separable metric space. We prove that $d i m_{T} X = i n f d i m_{L} X^{'} : X^{'} i s h o m e o m o r p h i c t o X$ , where $d i m_{L} X$ and $d i m_{T} X$ stand for the concentration dimension and the topological dimension of X, respectively.

A poset of topologies on the set of real numbers

Vitalij A. Chatyrko, Yasunao Hattori (2013)

Commentationes Mathematicae Universitatis Carolinae

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On the set $ℝ$ of real numbers we consider a poset $𝒫_{τ} (ℝ)$ (by inclusion) of topologies $τ (A)$ , where $A \subseteq ℝ$ , such that $A_{1} \supseteq A_{2}$ iff $τ (A_{1}) \subseteq τ (A_{2})$ . The poset has the minimal element $τ (ℝ)$ , the Euclidean topology, and the maximal element $τ (\emptyset)$ , the Sorgenfrey topology. We are interested when two topologies $τ_{1}$ and $τ_{2}$ (especially, for $τ_{2} = τ (\emptyset)$ ) from the poset define homeomorphic spaces $(ℝ, τ_{1})$ and $(ℝ, τ_{2})$ . In particular, we prove that for a closed subset $A$ of $ℝ$ the space $(ℝ, τ (A))$ is homeomorphic to the Sorgenfrey line $(ℝ, τ (\emptyset))$ iff $A$ is countable. We study also common...

Addition theorems for dense subspaces

Aleksander V. Arhangel'skii (2015)

Commentationes Mathematicae Universitatis Carolinae

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We study topological spaces that can be represented as the union of a finite collection of dense metrizable subspaces. The assumption that the subspaces are dense in the union plays a crucial role below. In particular, Example 3.1 shows that a paracompact space $X$ which is the union of two dense metrizable subspaces need not be a $p$ -space. However, if a normal space $X$ is the union of a finite family $μ$ of dense subspaces each of which is metrizable by a complete metric, then $X$ is also metrizable...

On dimensionally restricted maps

H. Murat Tuncali, Vesko Valov (2002)

Fundamenta Mathematicae

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Let f: X → Y be a closed n-dimensional surjective map of metrizable spaces. It is shown that if Y is a C-space, then: (1) the set of all maps g: X → ⁿ with dim(f △ g) = 0 is uniformly dense in C(X,ⁿ); (2) for every 0 ≤ k ≤ n-1 there exists an $F_{σ}$ -subset $A_{k}$ of X such that $d i m A_{k} \leq k$ and the restriction $f | (X ∖ A_{k})$ is (n-k-1)-dimensional. These are extensions of theorems by Pasynkov and Toruńczyk, respectively, obtained for finite-dimensional spaces. A generalization of a result due to Dranishnikov and Uspenskij...

Borsuk-Sieklucki theorem in cohomological dimension theory

Margareta Boege, Jerzy Dydak, Rolando Jiménez, Akira Koyama, Evgeny V. Shchepin (2002)

Fundamenta Mathematicae

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The Borsuk-Sieklucki theorem says that for every uncountable family ${X_{α}}_{α \in A}$ of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist α ≠ β such that $d i m (X_{α} \cap X_{β}) = n$ . In this paper we show a cohomological version of that theorem: Theorem. Suppose a compactum X is $c l c_{ℤ}^{n + 1}$ , where n ≥ 1, and G is an Abelian group. Let ${X_{α}}_{α \in J}$ be an uncountable family of closed subsets of X. If $d i m_{G} X = d i m_{G} X_{α} = n$ for all α ∈ J, then $d i m_{G} (X_{α} \cap X_{β}) = n$ for some α ≠ β. For G being a countable principal ideal domain the above result was proved by Choi...