Connected economically metrizable spaces

Taras Banakh; Myroslava Vovk; Michał Ryszard Wójcik

Displaying similar documents to “Connected economically metrizable spaces”

Characterizing metric spaces whose hyperspaces are homeomorphic to ℓ₂

T. Banakh, R. Voytsitskyy (2008)

Colloquium Mathematicae

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It is shown that the hyperspace $C l d_{H} (X)$ (resp. $B d d_{H} (X)$ ) of non-empty closed (resp. closed and bounded) subsets of a metric space (X,d) is homeomorphic to ℓ₂ if and only if the completion X̅ of X is connected and locally connected, X is topologically complete and nowhere locally compact, and each subset (resp. each bounded subset) of X is totally bounded.

On D-connected sets in the space $V^{q}$

S. Topa (1976)

Annales Polonici Mathematici

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On the tangency of sets in generalized metric spaces for certain functions of the class $F_{ρ}^{*}$

T. Konik (1991)

Matematički Vesnik

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On maps preserving connectedness and/or compactness

István Juhász, Jan van Mill (2018)

Commentationes Mathematicae Universitatis Carolinae

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We call a function $f : X \to Y$ P-preserving if, for every subspace $A \subset X$ with property P, its image $f (A)$ also has property P. Of course, all continuous maps are both compactness- and connectedness-preserving and the natural question about when the converse of this holds, i.e. under what conditions such a map is continuous, has a long history. Our main result is that any nontrivial product function, i.e. one having at least two nonconstant factors, that has connected domain, $T_{1}$ range, and is connectedness-preserving...

The nonexistence of universal metric flows

Stefan Geschke (2018)

Commentationes Mathematicae Universitatis Carolinae

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We consider dynamical systems of the form $(X, f)$ where $X$ is a compact metric space and $f : X \to X$ is either a continuous map or a homeomorphism and provide a new proof that there is no universal metric dynamical system of this kind. The same is true for metric minimal dynamical systems and for metric abstract $ω$ -limit sets, answering a question by Will Brian.

On the connectivity of finite subset spaces

Jacob Mostovoy, Rustam Sadykov (2012)

Fundamenta Mathematicae

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We prove that the space $e x p_{k} ⋁ S^{m + 1}$ of nonempty subsets of cardinality at most k in a bouquet of m+1-dimensional spheres is (m+k-2)-connected. This, as shown by Tuffley, implies that the space $e x p_{k} X$ is (m+k-2)-connected for any m-connected cell complex X.

Wijsman hyperspaces of non-separable metric spaces

Rodrigo Hernández-Gutiérrez, Paul J. Szeptycki (2015)

Fundamenta Mathematicae

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Given a metric space ⟨X,ρ⟩, consider its hyperspace of closed sets CL(X) with the Wijsman topology $τ_{W (ρ)}$ . It is known that $⟨ C L (X), τ_{W (ρ)} ⟩$ is metrizable if and only if X is separable, and it is an open question by Di Maio and Meccariello whether this is equivalent to $⟨ C L (X), τ_{W (ρ)} ⟩$ being normal. We prove that if the weight of X is a regular uncountable cardinal and X is locally separable, then $⟨ C L (X), τ_{W (ρ)} ⟩$ is not normal. We also solve some questions by Cao, Junnila and Moors regarding isolated points in Wijsman hyperspaces. ...

Isometric embeddings of a class of separable metric spaces into Banach spaces

Sophocles K. Mercourakis, Vassiliadis G. Vassiliadis (2018)

Commentationes Mathematicae Universitatis Carolinae

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Let $(M, d)$ be a bounded countable metric space and $c > 0$ a constant, such that $d (x, y) + d (y, z) - d (x, z) \geq c$ , for any pairwise distinct points $x, y, z$ of $M$ . For such metric spaces we prove that they can be isometrically embedded into any Banach space containing an isomorphic copy of $ℓ_{\infty}$ .

Domination numbers in graphs with removed edge or set of edges

Magdalena Lemańska (2005)

Discussiones Mathematicae Graph Theory

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It is known that the removal of an edge from a graph G cannot decrease a domination number γ(G) and can increase it by at most one. Thus we can write that γ(G) ≤ γ(G-e) ≤ γ(G)+1 when an arbitrary edge e is removed. Here we present similar inequalities for the weakly connected domination number $γ_{w}$ and the connected domination number $γ_{c}$ , i.e., we show that $γ_{w} (G) \leq γ_{w} (G - e) \leq γ_{w} (G) + 1$ and $γ_{c} (G) \leq γ_{c} (G - e) \leq γ_{c} (G) + 2$ if G and G-e are connected. Additionally we show that $γ_{w} (G) \leq γ_{w} (G - E ₚ) \leq γ_{w} (G) + p - 1$ and $γ_{c} (G) \leq γ_{c} (G - E ₚ) \leq γ_{c} (G) + 2 p - 2$ if G and G - Eₚ are connected and Eₚ = E(Hₚ) where Hₚ of order...

Metric enrichment, finite generation, and the path coreflection

Alexandru Chirvasitu (2024)

Archivum Mathematicum

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We prove a number of results involving categories enriched over CMet, the category of complete metric spaces with possibly infinite distances. The category CPMet of path complete metric spaces is locally $ℵ_{1}$ -presentable, closed monoidal, and coreflective in CMet. We also prove that the category CCMet of convex complete metric spaces is not closed monoidal and characterize the isometry- $ℵ_{0}$ -generated objects in CMet, CPMet and CCMet, answering questions by Di Liberti and Rosický. Other results...

Connected components of sets of finite perimeter and applications to image processing

Luigi Ambrosio, Vicent Caselles, Simon Masnou, Jean-Michel Morel (2001)

Journal of the European Mathematical Society

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This paper contains a systematic analysis of a natural measure theoretic notion of connectedness for sets of finite perimeter in $ℝ^{N}$ , introduced by H. Federer in the more general framework of the theory of currents. We provide a new and simpler proof of the existence and uniqueness of the decomposition into the so-called $M$ -connected components. Moreover, we study carefully the structure of the essential boundary of these components and give in particular a reconstruction formula of a set...

Geometry of the Banach spaces C(βℕ × K,X) for compact metric spaces K

Dale E. Alspach, Elói Medina Galego (2011)

Studia Mathematica

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A classical result of Cembranos and Freniche states that the C(K,X) space contains a complemented copy of c₀ whenever K is an infinite compact Hausdorff space and X is an infinite-dimensional Banach space. This paper takes this result as a starting point and begins a study of conditions under which the spaces C(α), α < ω₁, are quotients of or complemented in C(K,X). In contrast to the c₀ result, we prove that if C(βℕ ×[1,ω],X) contains a complemented copy of $C (ω^{ω})$ then X contains a copy...

Adjacent dyadic systems and the $L^{p}$ -boundedness of shift operators in metric spaces revisited

Olli Tapiola (2016)

Colloquium Mathematicae

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With the help of recent adjacent dyadic constructions by Hytönen and the author, we give an alternative proof of results of Lechner, Müller and Passenbrunner about the $L^{p}$ -boundedness of shift operators acting on functions $f \in L^{p} (X; E)$ where 1 < p < ∞, X is a metric space and E is a UMD space.

The (dis)connectedness of products of Hausdorff spaces in the box topology

Vitalij A. Chatyrko (2021)

Commentationes Mathematicae Universitatis Carolinae

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In this paper the following two propositions are proved: (a) If $X_{α}$ , $α \in A$ , is an infinite system of connected spaces such that infinitely many of them are nondegenerated completely Hausdorff topological spaces then the box product $□_{α \in A} X_{α}$ can be decomposed into continuum many disjoint nonempty open subsets, in particular, it is disconnected. (b) If $X_{α}$ , $α \in A$ , is an infinite system of Brown Hausdorff topological spaces then the box product $□_{α \in A} X_{α}$ is also Brown Hausdorff, and hence, it is connected. A space...

Does the endomorphism poset $P^{P}$ determine whether a finite poset $P$ is connected? An issue Duffus raised in 1978

Jonathan David Farley (2023)

Mathematica Bohemica

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Duffus wrote in his 1978 Ph.D. thesis, “It is not obvious that $P$ is connected and $P^{P} ≅ Q^{Q}$ imply that $Q$ is connected”, where $P$ and $Q$ are finite nonempty posets. We show that, indeed, under these hypotheses $Q$ is connected and $P ≅ Q$ .

Some approximate fixed point theorems without continuity of the operator using auxiliary functions

Sumit Chandok, Arslan Hojjat Ansari, Tulsi Dass Narang (2019)

Mathematica Bohemica

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We introduce partial generalized convex contractions of order $4$ and rank $4$ using some auxiliary functions. We present some results on approximate fixed points and fixed points for such class of mappings having no continuity condition in $α$ -complete metric spaces and $μ$ -complete metric spaces. Also, as an application, some fixed point results in a metric space endowed with a binary relation and some approximate fixed point results in a metric space endowed with a graph have been obtained....

The discriminant and oscillation lengths for contact and Legendrian isotopies

Vincent Colin, Sheila Sandon (2015)

Journal of the European Mathematical Society

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We define an integer-valued non-degenerate bi-invariant metric (the discriminant metric) on the universal cover of the identity component of the contactomorphism group of any contact manifold. This metric has a very simple geometric definition, based on the notion of discriminant points of contactomorphisms. Using generating functions we prove that the discriminant metric is unbounded for the standard contact structures on $ℝ^{2 n} \times S^{1}$ and $ℝ P^{2 n + 1}$ . On the other hand we also show by elementary arguments...