Further new generalized topologies via mixed constructions due to Császár

Erdal Ekici

Displaying similar documents to “Further new generalized topologies via mixed constructions due to Császár”

A hit-and-miss topology for $2^{X}$ , Cₙ(X) and Fₙ(X)

Benjamín Espinoza, Verónica Martínez-de-la-Vega, Jorge M. Martínez-Montejano (2009)

Colloquium Mathematicae

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A hit-and-miss topology ( $τ_{H M}$ ) is defined for the hyperspaces $2^{X}$ , Cₙ(X) and Fₙ(X) of a continuum X. We study the relationship between $τ_{H M}$ and the Vietoris topology and we find conditions on X for which these topologies are equivalent.

Connectedness of some rings of quotients of $C (X)$ with the $m$ -topology

F. Azarpanah, M. Paimann, A. R. Salehi (2015)

Commentationes Mathematicae Universitatis Carolinae

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In this article we define the $m$ -topology on some rings of quotients of $C (X)$ . Using this, we equip the classical ring of quotients $q (X)$ of $C (X)$ with the $m$ -topology and we show that $C (X)$ with the $r$ -topology is in fact a subspace of $q (X)$ with the $m$ -topology. Characterization of the components of rings of quotients of $C (X)$ is given and using this, it turns out that $q (X)$ with the $m$ -topology is connected if and only if $X$ is a pseudocompact almost $P$ -space, if and only if $C (X)$ with $r$ -topology is connected. We also...

The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

S. Gabriyelyan, J. Kąkol, G. Plebanek (2016)

Studia Mathematica

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Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset of $C_{k} (X)$ is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every $k_{ℝ}$ -space, hence any k-space, is Ascoli. Let X be a metrizable space. We prove that the space $C_{k} (X)$ is Ascoli iff $C_{k} (X)$ is a $k_{ℝ}$ -space iff X is locally compact. Moreover, $C_{k} (X)$ endowed with the weak topology is Ascoli iff X is countable and discrete. Using some basic concepts from probability...

Order intervals in $C (K)$ . Compactness, coincidence of topologies, metrizability

Zbigniew Lipecki (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let $K$ be a compact space and let $C (K)$ be the Banach lattice of real-valued continuous functions on $K$ . We establish eleven conditions equivalent to the strong compactness of the order interval $[0, x]$ in $C (K)$ , including the following ones: (i) ${x > 0}$ consists of isolated points of $K$ ; (ii) $[0, x]$ is pointwise compact; (iii) $[0, x]$ is weakly compact; (iv) the strong topology and that of pointwise convergence coincide on $[0, x]$ ; (v) the strong and weak topologies coincide on $[0, x]$ . Moreover, the weak topology and that of pointwise...

The common division topology on $ℕ$

José del Carmen Alberto-Domínguez, Gerardo Acosta, Maira Madriz-Mendoza (2022)

Commentationes Mathematicae Universitatis Carolinae

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A topological space $X$ is totally Brown if for each $n \in ℕ ∖ {1}$ and every nonempty open subsets $U_{1}, U_{2}, ..., U_{n}$ of $X$ we have ${cl}_{X} (U_{1}) \cap {cl}_{X} (U_{2}) \cap \dots \cap {cl}_{X} (U_{n}) \neq \emptyset$ . Totally Brown spaces are connected. In this paper we consider a topology $τ_{S}$ on the set $ℕ$ of natural numbers. We then present properties of the topological space $(ℕ, τ_{S})$ , some of them involve the closure of a set with respect to this topology, while others describe subsets which are either totally Brown or totally separated. Our theorems generalize results proved by P. Szczuka in 2013, 2014,...

A compact Hausdorff topology that is a T₁-complement of itself

Dmitri Shakhmatov, Michael Tkachenko (2002)

Fundamenta Mathematicae

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Topologies τ₁ and τ₂ on a set X are called T₁-complementary if τ₁ ∩ τ₂ = X∖F: F ⊆ X is finite ∪ ∅ and τ₁∪τ₂ is a subbase for the discrete topology on X. Topological spaces $(X, τ_{X})$ and $(Y, τ_{Y})$ are called T₁-complementary provided that there exists a bijection f: X → Y such that $τ_{X}$ and $f^{- 1} (U) : U \in τ_{Y}$ are T₁-complementary topologies on X. We provide an example of a compact Hausdorff space of size $2^{}$ which is T₁-complementary to itself ( denotes the cardinality of the continuum). We prove that the existence of a compact...

The Golomb space is topologically rigid

Taras O. Banakh, Dario Spirito, Sławomir Turek (2021)

Commentationes Mathematicae Universitatis Carolinae

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The Golomb space $ℕ_{τ}$ is the set $ℕ$ of positive integers endowed with the topology $τ$ generated by the base consisting of arithmetic progressions ${a + b n : n \geq 0}$ with coprime $a, b$ . We prove that the Golomb space $ℕ_{τ}$ is topologically rigid in the sense that its homeomorphism group is trivial. This resolves a problem posed by T. Banakh at Mathoverflow in 2017.

Interaction between cellularity of Alexandroff spaces and entropy of generalized shift maps

Fatemah Ayatollah Zadeh Shirazi, Sahar Karimzadeh Dolatabad, Sara Shamloo (2016)

Commentationes Mathematicae Universitatis Carolinae

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In the following text for a discrete finite nonempty set $K$ and a self-map $ϕ : X \to X$ we investigate interaction between different entropies of generalized shift $σ_{ϕ} : K^{X} \to K^{X}$ , ${(x_{α})}_{α \in X} \mapsto {(x_{ϕ (α)})}_{α \in X}$ and cellularities of some Alexandroff topologies on $X$ .

On the existence of exactly one solution of integral equations in the space $L^{P}$ with a mixed norm

Bogdan Rzepecki (1975)

Annales Polonici Mathematici

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Totally Brown subsets of the Golomb space and the Kirch space

José del Carmen Alberto-Domínguez, Gerardo Acosta, Gerardo Delgadillo-Piñón (2022)

Commentationes Mathematicae Universitatis Carolinae

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A topological space $X$ is totally Brown if for each $n \in ℕ ∖ {1}$ and every nonempty open subsets $U_{1}, U_{2}, ..., U_{n}$ of $X$ we have ${cl}_{X} (U_{1}) \cap {cl}_{X} (U_{2}) \cap \dots \cap {cl}_{X} (U_{n}) \neq \emptyset$ . Totally Brown spaces are connected. In this paper we consider the Golomb topology $τ_{G}$ on the set $ℕ$ of natural numbers, as well as the Kirch topology $τ_{K}$ on $ℕ$ . Then we examine subsets of these spaces which are totally Brown. Among other results, we characterize the arithmetic progressions which are either totally Brown or totally separated in $(ℕ, τ_{G})$ . We also show that $(ℕ, τ_{G})$ and $(ℕ, τ_{K})$ are aposyndetic....

The Spaces of Closed Convex Sets in Euclidean Spaces with the Fell Topology

Katsuro Sakai, Zhongqiang Yang (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let $C o n v_{F} (ℝ ⁿ)$ be the space of all non-empty closed convex sets in Euclidean space ℝ ⁿ endowed with the Fell topology. We prove that $C o n v_{F} (ℝ ⁿ) \approx ℝ ⁿ \times Q$ for every n > 1 whereas $C o n v_{F} (ℝ) \approx ℝ \times$ .

Topological properties of some spaces of continuous operators

Marian Nowak (2016)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let $C_{b} (X, E)$ be the space of all E-valued bounded continuous functions on X, equipped with the strict topology β. We study topological properties of the space $L_{β} (C_{b} (X, E), F)$ of all $(β, | | \cdot | |_{F})$ -continuous linear operators from $C_{b} (X, E)$ to F, equipped with the topology $τ_{s}$ of simple convergence. If X is a locally compact paracompact space (resp. a P-space), we characterize $τ_{s}$ -compact subsets of $L_{β} (C_{b} (X, E), F)$ in terms of properties of the corresponding sets of the representing...