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Let $ (G, X) $ (G , X) be a G-system, which means that X is a compact Hausdorff space and G is an infinite topological group continuously acting on X, and let μ be a G-invariant measure of $ (G, X) $ (G , X). In this paper, we introduce the concepts of rigidity, uniform rigidity and μ-Ω-equicontinuity of $ (G,X) $ (G , X) with respect to an infinite sequence Ω of G and the notions of μ-Ω-equicontinuity and μ-Ω-mean-equicontinuity of a function $ f\in L^2(\mu) $ f ∈ L 2 (μ) with respect to an infinite sequence Ω of G. Then we give some equivalent conditions for $ f\in L^2(\mu) $ f ∈ L 2 (μ) and $ (G,X) $ (G , X) to be rigid, respectively. In addition, if G is commutative and X satisfies the first axiom of countability, we present some equivalent conditions for $ (G,X) $ (G , X) to be uniformly rigid. [ABSTRACT FROM AUTHOR]

We characterize subhomogeneity for twisted étale groupoid |$\text{C}^{*}$| -algebras and obtain an upper bound on their nuclear dimension. As an application, we remove the principality assumption in recent results on upper bounds on the nuclear dimension of a twisted étale groupoid |$\text{C}^{*}$| -algebra in terms of the dynamic asymptotic dimension of the groupoid and the covering dimension of its unit space. As a non-principal example, we show that the dynamic asymptotic dimension of any minimal (not necessarily free) action of the infinite dihedral group |$D_{\infty }$| on an infinite compact Hausdorff space |$X$| is always one. So if we further assume that |$X$| is second-countable and has finite covering dimension, then |$C(X)\rtimes _{r} D_{\infty }$| has finite nuclear dimension and is classifiable by its Elliott invariant. [ABSTRACT FROM AUTHOR]

We deal with the class of Hausdorff spaces having a π -base whose elements have an H-closed closure. Carlson proved that | X | ≤ 2 w L (X) ψ c (X) t (X) for every quasiregular space X with a π -base whose elements have an H-closed closure. We provide an example of a space X having a π -base whose elements have an H-closed closure which is not quasiregular (neither Urysohn) such that | X | > 2 w L (X) χ (X) (then | X | > 2 w L (X) ψ c (X) t (X) ). Always in the class of spaces with a π -base whose elements have an H-closed closure, we establish the bound | X | ≤ 2 w L (X) k (X) for Urysohn spaces and we give an example of an Urysohn space Z such that k (Z) < χ (Z) . Lastly, we present some equivalent conditions to the Martin's Axiom involving spaces with a π -base whose elements have an H-closed closure and, additionally, we prove that if a quasiregular space has a π -base whose elements have an H-closed closure then such a space is Baire. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of the α -fractal function and fractal approximation for a set-valued continuous map defined on a closed and bounded interval of real numbers. Also, we study some properties of such fractal functions. Further, we estimate the perturbation error between the given continuous function and its α -fractal function. Additionally, we define a new graph of a set-valued function different from the standard graph introduced in the literature and establish some bounds on the fractal dimension of the newly defined graph of some special classes of set-valued functions. Also, we explain the need to define this new graph with examples. In the sequel, we prove that this new graph of an α -fractal function is an attractor of an iterated function system. [ABSTRACT FROM AUTHOR]

For the set I (X) of probability measures on a compact Hausdorff space X, we propose a new way to introduce the topology by using the open subsets of the space X. Then, among other things, we give a new proof that for a compact Hausdorff space X, the space I (X) is also a compact Hausdorff space. For a Tychonoff space X, we consider the topological space I τ (X) of τ -smooth idempotent probability measures on X and show that the space I τ (X) is Čech-complete if and only if the given space X is Čech-complete. [ABSTRACT FROM AUTHOR]

The symbol TD(X) denotes the hyperspace of all nonempty totally disconnected compact subsets of a Hausdorff space X. This hyperspace is endowed with the Vietoris topology. For a mapping between Hausdorff spaces f : X → Y, define the induced mapping TD(f) : TD(X) → TD(Y) by TD(f)(A) = f(A) (the image of A under f). In the current paper, we study the relationships between the condition f belongs to a class of mappings between Hausdorff spaces 필 and the condition TD(f) belongs to 필. [ABSTRACT FROM AUTHOR]

The N-star compactifications of frames are the frame-theoretic counterpart of the N-point compactifications of locally compact Hausdorff spaces. A π -compactification of a frame L is a compactification constructed using a special type of a basis called a π -compact basis; the Freudenthal compactification is the largest π -compactification of a rim-compact frame. As one of the main results, we show that the Freudenthal compactification of a regular continuous frame is the least upper bound for the set of all N-star compactifications. A compactification whose right adjoint preserves disjoint binary joins is called perfect. We establish a class of frames for which N-star compactifications are always perfect. For the class of zero-dimensional frames, we construct a compactification which is isomorphic to the Banaschewski compactification and the Freudenthal compactification; in some special case, this compactification is isomorphic to the Stone–Čech compactification. [ABSTRACT FROM AUTHOR]

In the present paper, we have obtained common best proximity point theorems of nonself maps in Hausdorff topological space. Further, our results extend the results due to Gerald F. Jungck, thereby proving a generalized version of Kirk's theorem (J. London Math. 1(1):107–111, 1969). [ABSTRACT FROM AUTHOR]

In this paper, by considering the notion of L-algebras and CL-algebras, we construct a topology on a bounded CL-algebra. Then, we investigate some of its topological properties, such as Hausdorff space, T 0 -space and T 1 -space and connectedness. Moreover, we express the relation between closed and compact sets in this topology. Finally, by considering the notion of semi-topological algebra, we prove that any commutative bounded CL-algebra is a right semi-topological algebra. [ABSTRACT FROM AUTHOR]

Let X be a completely regular Hausdorff space, let E and F denote Banach spaces. Let C_b(X,E) denote the space of E-valued bounded continuous functions on X and let \beta be the strict topology on this space. We establish the relationship between nuclear operators T:C_b(X,E)\rightarrow F between the locally convex space (C_b(X,E),\beta) and the Banach space F and their representing operator-valued Borel measures. [ABSTRACT FROM AUTHOR]

Existing line segment detectors break perceptually contiguous linear structures into multiple line segments. This can be offset by re-merging the segments, but existing merging algorithms over-merge and produce globally incorrect segments. Geometric cues are necessary but not sufficient for deciding whether to merge two segments or not. By restricting the result of any merging decision to have underlying image support, we reduce over-merging and globally incorrect segments. We propose a novel measure for evaluating merged segments based on line segment Hausdorff distance. On images from YorkUrbanDB, we show that our algorithm improves both qualitative and quantitative results obtained from four existing line segment detection methods and is better than two existing line segment merging methods. Our method does not suffer from inconsistent results produced by four recent deep learning-based models. The method is easily customisable to work for line drawings such as hand-drawn maps to obtain vectorized representations. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study prime spectrums of EQ-algebras and to solve two open problems about |$\wedge $| -prime spectrums of involutive and prelinear EQ-algebras, which were proposed by N. Akhlaghinia, R.A. Borzooei and M. A. Kologani. In order to do so, we first give some characterizations of preideals, prime preideals and maximal preideals on (good) EQ-algebras, respectively. Then we introduce the notion of quasi De Morgan EQ-algebras (MEQ-algebras for short) and obtain that |$\wedge $| -prime preideals coincide with prime preideals for MEQ-algebras, and each involutive EQ-algebra is an MEQ-algebra. Following, we show that the prime spectrum space of a good EQ-algebra is a compact topological space and obtain that for any involutive EQ-algebra the prime spectrum space is connected if and only if its Boolean center is indeed 2-element. Also, we prove that the maximal spectrum space of a good and prelinear EQ-algebra (or an involutive and prelinear EQ-algebra) is a normal Hausdorff space. These results totally answer the above two open problems. Finally, we give some characterizations of the spectrum space of an MEQ-algebra by its reticulation. [ABSTRACT FROM AUTHOR]

We give necessary and sufficient conditions on non-negative weights u, v and measures μ, v in the inequality (∫Ω |T f(x)|qu(x)d μ(x)1/q ≤ C (∫Ω |f(x)|p v(x)dv(x)))1/p. Here the integral operator T is a Hardy-Steklov type operator associated with a family of open subsets Ω(t) of an open set Ω in a Hausdorff topological space X; μ,v are σ-additive Borel measures, and 1 < p < ∞, 0 < q < ∞. The integration in T is over domains of type Ω(b(t))\Ω(a(t)) where a, b are non-negative, increasing, continuous functions on [0,∞) that vanish at zero, tend to ∞ at ∞ and satisfy a (t) < b (t) for t ∈ (0, ∞). Previously such results have been known for an operator on a subset of a Euclidean space. [ABSTRACT FROM AUTHOR]

For a free filter F on ω , endow the space N F = ω ∪ { p F } , where p F ∉ ω , with the topology in which every element of ω is isolated whereas all open neighborhoods of p F are of the form A ∪ { p F } for A ∈ F . Spaces of the form N F constitute the class of the simplest non-discrete Tychonoff spaces. The aim of this paper is to study them in the context of the celebrated Josefson–Nissenzweig theorem from Banach space theory. We prove, e.g., that, for a filter F, the space N F carries a sequence ⟨ μ n : n ∈ ω ⟩ of normalized finitely supported signed measures such that μ n (f) → 0 for every bounded continuous real-valued function f on N F if and only if F ∗ ≤ K Z , that is, the dual ideal F ∗ is Katětov below the asymptotic density ideal Z . Consequently, we get that if F ∗ ≤ K Z , then: (1) if X is a Tychonoff space and N F is homeomorphic to a subspace of X, then the space C p ∗ (X) of bounded continuous real-valued functions on X contains a complemented copy of the space c 0 endowed with the pointwise topology, (2) if K is a compact Hausdorff space and N F is homeomorphic to a subspace of K, then the Banach space C(K) of continuous real-valued functions on K is not a Grothendieck space. The latter result generalizes the well-known fact stating that if a compact Hausdorff space K contains a non-trivial convergent sequence, then the space C(K) is not Grothendieck. [ABSTRACT FROM AUTHOR]

One point compactification is studied in the light of ideal of subsets of ℕ. 퓘-proper map is introduced and showed that a continuous map can be extended continuously to the one point 퓘-compactification if and only if the map is 퓘-proper. Nowhere tallness, introduced by P. Matet and J. Pawlikowski in [J. Symb. Log. 63(3) (1998), 1040–1054], plays an important role in this article to study various properties of 퓘-proper maps. It is seen that one point 퓘-compactification of a topological space may fail to be Hausdorff even if the underlying topological space is Hausdorff but a class {퓘} of ideals has been identified for which one point 퓘-compactification coincides with the one point compactification if the underlying topological space is metrizable. Let's speak our minds that the results in this article will look elegant if one looks at it from a topological angle. [ABSTRACT FROM AUTHOR]

Let $ (X, f) $ (X , f) be a dynamical system, i.e. X is a compact metric space and f is a continuous self-map on X and let $ K(X) $ K (X) , $ M(X) $ M (X) and $ \mathbb {F}(X) $ F (X) be the sets of all non-empty compact subsets of X, Borel probability measures on X and upper semi-continuous fuzzy sets on X, respectively. Then $ K(X) $ K (X) , $ M(X) $ M (X) and $ \mathbb {F}(X) $ F (X) are metric spaces under Hausdorff metric, prohorov metric and level-wise metric, respectively. Therefore, $ (X, f) $ (X , f) naturally induces three new systems $ (K(X), \bar {f}) $ (K (X) , f ¯) , $ (M(X), \hat {f}) $ (M (X) , f ^) and $ (\mathbb {F}(X), \tilde {f}) $ (F (X) , f ~). In this article, we investigate the connection of $ (r,s) $ (r , s) -sensitivity, $ (r,s) $ (r , s) -asymptotic sensitivity, $ (r,s) $ (r , s) -Li-Yorke sensitivity and Δ-transitivity of $ (X, f) $ (X , f) and its induced systems $ (K(X),\bar {f}) $ (K (X) , f ¯) , $ (M(X),\hat {f}) $ (M (X) , f ^) and $ (\mathbb {F}(X),\tilde {f}) $ (F (X) , f ~) and we obtain some desired results. For instance, we prove that $ (K(X),\bar {f}) $ (K (X) , f ¯) is $ (r,s) $ (r , s) -sensitive ⇔ $ (\mathbb {F}^{1}(X),\widetilde {f_{g}}) $ (F 1 (X) , f g ~) is $ (r,s) $ (r , s) -sensitive for each $ g\in D_{m}(I) $ g ∈ D m (I) satisfying $ g^{-1}(1)=\{1\} $ g − 1 (1) = { 1 } ; $ (X,f) $ (X , f) is Δ-transitive ⇔ $ (K(X),\bar {f}) $ (K (X) , f ¯) is Δ-transitive ⇔ $ (M(X),\hat {f}) $ (M (X) , f ^) is Δ-transitive $ \Leftrightarrow \,(\mathbb {F}^{1}(X),\tilde {f}) $ ⇔ (F 1 (X) , f ~) is Δ-transitive. [ABSTRACT FROM AUTHOR]

Assume that M is a transitive model of $ZFC+CH$ containing a simplified $(\omega _1,2)$ -morass, $P\in M$ is the poset adding $\aleph _3$ generic reals and G is P -generic over M. In M we construct a function between sets of terms in the forcing language, that interpreted in $M[G]$ is an $\mathbb R$ -linear order-preserving monomorphism from the finite elements of an ultrapower of the reals, over a non-principal ultrafilter on $\omega $ , into the Esterle algebra of formal power series. Therefore it is consistent that $2^{\aleph _0}>\aleph _2$ and, for any infinite compact Hausdorff space X , there exists a discontinuous homomorphism of $C(X)$ , the algebra of continuous real-valued functions on X. [ABSTRACT FROM AUTHOR]

As usual denote by ℓ22$\ell _2^2$ the real two‐dimensional Hilbert space. We prove that if K$K$ and S$S$ are locally compact Hausdorff spaces and T$T$ is a linear isomorphism from C0(K,ℓ22)$C_{0}(K,\ell _2^2)$ onto C0(S,ℓ22)$C_{0}(S,\ell _2^2)$ satisfying ∥T∥∥T−1∥≤2.054208,$$\begin{equation*} \hspace*{115pt}{\Vert T\Vert} \ {\Vert T^{-1}\Vert} \le \sqrt {2.054208}, \end{equation*}$$then K$K$ and S$S$ are homeomorphic. This theorem is the strongest of all the other vector‐valued Banach–Stone theorems known so far in the sense that in none of them the distortion of the isomorphism T$T$, denoted by ∥T∥∥T−1∥${\Vert T\Vert} \ {\Vert T^{-1}\Vert}$, is as large as 2.054208$\sqrt {2.054208}$. Some remarks on the proof method developed here to prove our theorem suggest the conjecture that it is in fact very close to the optimal Banach–Stone theorem for C0(K,ℓ22)$C_{0}(K, \ell _2^2)$ spaces, or in more precise words, the exact value of the Banach–Stone constant of ℓ22$\ell _2^2$ is between 2.054208$\sqrt {2.054208}$ and 2.054209$\sqrt {2.054209}$. [ABSTRACT FROM AUTHOR]

In this article, we introduce the notions of l-transfer lower continuous and q-level intersectionally closed for set-valued mappings with respect to the lower set less relation. Then, we obtain some existence results for strict weak l-efficient solutions of such set-valued mappings. Moreover, we prove some existence results for nonconvex set optimization problems via asymptotic analysis tools, in the setting of the Banach spaces equipped with a Hausdorff topology σ coarser than the norm topology. [ABSTRACT FROM AUTHOR]

For a Hausdorff non-compact space X, relationships between closure-type properties of the hyperspace (Λ,τ+Δ) and covering properties of that of X have been studied. We then investigate selective separability and some variations of this property. Finally supertightness of (Λ,τ+Δ) has been studied. [ABSTRACT FROM AUTHOR]

Let G = (V (G), E(G)) be a simple undirected graph. A block of G is a maximal connected subgraph of G that contains no cut-vertices [11]. The family of vertex sets of blocks of G generates a unique topology. In this paper, we formally define the topology generated by the family of blocks in a graph called the block topological space. Moreover, we characterize and describe some special attributes of the block topological space. Finally, we associate a corresponding graph from a given block topological space by defining the block topological graph. [ABSTRACT FROM AUTHOR]

We give a pointwise version of sensitivity in terms of open covers for a semiflow (T, X) of a topological semigroup T on a Hausdorff space X and call it a Hausdorff sensitive point. If (X , U) is a uniform space with topology τ , then the definition of Hausdorff sensitivity for (T , (X , τ)) gives a pointwise version of sensitivity in terms of uniformity and we call it a uniformly sensitive point. For a semiflow (T, X) on a compact Hausdorff space X, these notions (i.e. Hausdorff sensitive point and uniformly sensitive point) are equal and they are T-invariant if T is a C-semigroup. They are not preserved by factor maps and subsystems, but behave slightly better with respect to lifting. We give the definition of a topologically equicontinuous pair for a semiflow (T, X) on a topological space X and show that if (T, X) is a topologically equicontinuous pair in (x, y), for all y ∈ X , then Tx ¯ = D T (x) where D T (x) = ⋂ { TU ¯ : for all open neighborhoods U of x }. We prove for a topologically transitive semiflow (T, X) of a C-semigroup T on a regular space X with a topologically equicontinuous point that the set of topologically equicontinuous points coincides with the set of transitive points. This implies that every minimal semiflow of C-semigroup T on a regular space X with a topologically equicontinuous point is topologically equicontinuous. Moreover, we show that if X is a regular space and (T, X) is not a topologically equicontinuous pair in (x, y), then x is a Hausdorff sensitive point for (T, X). Hence, a minimal semiflow of a C-semigroup T on a regular space X is either topologically equicontinuous or topologically sensitive. [ABSTRACT FROM AUTHOR]

Copyright of Cahiers de Topologie et Geometrie Differentielle Categoriques is the property of Andree C. EHRESMANN and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Cahiers de Topologie et Geometrie Differentielle Categoriques is the property of Andree C. EHRESMANN and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Suppose $ X $ is a compact connected metric space and $ f\colon X \to X $ is a metric coarse expanding conformal map in the sense of Haïssinsky-Pilgrim. We show that if $ X $ contains a homeomorphic copy of the letter 'Y', then the Hausdorff dimension of $ X $ is greater than one. As an application, we show that for a semi-hyperbolic rational map $ f $ its Julia set $ \mathcal{J}_f $ is quasi-symmetric equivalent to a space having Hausdorff dimension 1 if and only if $ \mathcal{J}_f $ is homeomorphic to a circle or a closed interval. [ABSTRACT FROM AUTHOR]

In the present paper, we obtain the results on the degree of convergence of an H$$ H $$‐periodic continuous function in generalized Hölder spaces using Hausdorff operator with monotonically non‐decreasing and monotonically non‐increasing rows of its hexagonal Fourier series. Some important corollaries are also deduced from our main theorems. Applications of main theorems are also obtained in this paper. [ABSTRACT FROM AUTHOR]

Chipman contended, in stark contrast to the conventional view, that, utility is not a real number but a vector, and that it is inherently lexicographic in nature. On the other hand, in recent years continuous multi-utility representations of a preorder on a topological space, which proved to be the best kind of continuous representation, have been deeply studied. In this paper, we first state a general result, which guarantees, for every preordered topological space, the existence of a lexicographic order-embedding of the Chipman type. Then, we combine the Chipman approach and the continuous multi-utility approach, by stating a theorem that guarantees, under certain general conditions, the coexistence of these two kinds of continuous representations. [ABSTRACT FROM AUTHOR]

It is proven that if X is a real strictly convex 2-dimensional space, then there exists \delta >0 such that if K and S are locally compact Hausdorff spaces and T is an isomorphism from C_{0}(K, X) onto C_{0}(S, X) satisfying \begin{equation*} \|T\| \ \|T^{-1}\| \leq \lambda (X)+ \delta, \end{equation*} then K and S are homeomorphic. Here \lambda (X) is the Schäffer constant of X given by \begin{equation*} \lambda (X)=\inf \{\max \{\|x-y\|,\|x+y\|\}\,:\,\|x\|= 1 \text { and } \|y\|= 1\}. \end{equation*} Even for the classical cases X=\ell _p^2, 1

The concept of q-rung orthopair neutrosophic set is introduced in this paper and fundamental properties of it are studied. Also the ordinary notion of topological space is extended to q-rung orthopair neutrosophic environment, as well as the fundamental concepts of convergence, continuity, compactness and Hausdorff topological space. All these generalizations are illustrated by suitable examples. [ABSTRACT FROM AUTHOR]

Let L be a truncated Archimedean vector lattice whose truncation is denoted by *. In a recent paper, we proved that there exists a locally compact Hausdorff space X such that L is a lattice isomorphic with a truncated vector lattice of functions in C∞(X) whose truncation is provided by meet with some characteristic function on X. This representation, no matter how interesting it is, has a major drawback, namely, C∞(X) need not be a vector lattice, unless X is extremally disconnected. The main purpose of this paper is to remedy this shortcoming by proving that, indeed, an extremally disconnected locally compact space X can be found such that L can be seen as an order dense vector sublattice of C∞(X) whose truncation is provided, again, by meet with some characteristic function on X. [ABSTRACT FROM AUTHOR]

It is commonly known that some topological spaces include structures that may be used to expand abstract notions. Primal structure is one such sort of structure. We provided the primal Hausdorff class of spaces, which included the class of all Hausdorff spaces. Furthermore, we provide the concepts of primal regular spaces and primal normal spaces. We present new theorems and results. [ABSTRACT FROM AUTHOR]

The goal of this paper is to establish a general fixed point theorem for compact single-valued continuous mappings in Hausdorff p-vector spaces, and a fixed point theorem for upper semicontinuous set-valued mappings in locally p-convex spaces for p 2 (0; 1]. These results not only provide a solution to Schauder conjecture in the affirmative under the setting of p-vector spaces for compact single-valued continuous mappings, but also show the existence of fixed points for upper semicontinuous set-valued mappings defined on s-convex subsets in Hausdorff locally p-convex spaces, which would be fundamental for nonlinear functional analysis, where s; p 2 (0; 1]. [ABSTRACT FROM AUTHOR]

We consider the large class G of locally convex spaces that includes, among others, the classes of (D F) -spaces and (L F) -spaces. For a space E in class G we have characterized that a subspace Y of (E , σ (E , E ′)) , endowed with the induced topology, is analytic if and only if Y has a σ (E , E ′) -compact resolution and is contained in a σ (E , E ′) -separable subset of E. This result is applied to reprove a known important result (due to Cascales and Orihuela) about weak metrizability of weakly compact sets in spaces of class G. The mentioned characterization follows from the following analogous result: The space C (X) of continuous real-valued functions on a completely regular Hausdorff space X endowed with a topology ξ stronger or equal than the pointwise topology τ p of C (X) is analytic iff (C (X) , ξ) is separable and is covered by a compact resolution. [ABSTRACT FROM AUTHOR]

We show that for a nice partial action n with closed domain of a compact group G on a profinite space X the orbit space X/ ~G is profinite, this leads to the fact that when G is profinite the enveloping space XG is also profinite. Moreover, we provide conditions for the induced quotient map πG: X → ~G of η to have a continuous section. Relations between continuous sections of πG and continuous sections of the quotient map induced by the enveloping action of η are also considered. At the end of this work, we prove that the category of separately continuous actions on profinite spaces is reflective in the category of separately continuous actions on compact Hausdorff spaces. [ABSTRACT FROM AUTHOR]

Let X and Y be compact Hausdorff spaces with Y hyperstonean. In this paper, we prove that if C(X, R) and C(Y, R) are lattice isomorphic then these Banach spaces are linearly isometric, and, consequently, X and Y are homeomorphic, which in turn implies that X is also hyperstonean. Actually, we prove more than what is ann [ABSTRACT FROM AUTHOR]

We consider the problem of characterization of topological spaces embedded into countably compact Hausdorff topological spaces. We study the separation axioms for subspaces of Hausdorff countably compact topological spaces and construct an example of a regular separable scattered topological space that cannot be embedded into an Urysohn countably compact topological space. [ABSTRACT FROM AUTHOR]

This paper introduces the notion of multi-sensitivity with respect to a vector within the context of non-autonomous dynamical systems on uniform spaces and provides insightful results regarding  N -sensitivity and strongly multi-sensitivity, along with their behaviors under various conditions. The main results established are as follows: (1) For a k-periodic nonautonomous dynamical system on a Hausdorff uniform space  (S , U) , the system  (S , f k ∘ ⋯ ∘ f 1)  exhibits  N -sensitivity (or strongly multi-sensitivity) if and only if the system  (S , f 1 , ∞)  displays  N -sensitivity (or strongly multi-sensitivity). (2) Consider a finitely generated family of surjective maps on uniform space  (S , U) . If the system  (S , f 1 , ∞)  is  N -sensitive, then the system  (S , f k , ∞)  is also  N -sensitive. When the family  f 1 , ∞  is feebly open, the converse statement holds true as well. (3) Within a finitely generated family on uniform space  (S , U) ,  N -sensitivity (and strongly multi-sensitivity) persists under iteration. (4) We present a sufficient condition under which an nonautonomous dynamical system on infinite Hausdorff uniform space demonstrates  N -sensitivity. [ABSTRACT FROM AUTHOR]

Given a topological space X, let UX and ηX: X → UX denote, respectively, the Salbany compactification of X and the compactification map called the Salbany map of X. For every continuous function f: X → Y, there is a continuous function Uf: UX → UY, called the Salbany lift of f, satisfying (Uf) ◦ ηX = ηY ◦ f. If a continuous function f: X → Y has a stably compact codomain Y, then there is a Salbany extension F: UX → Y of f, not necessarily unique, such that F ◦ ηX = f. In this paper, we give a condition on a space such that its Salbany map is open. In particular, we prove that in a class of Hausdorff spaces, the spaces with open Salbany maps are precisely those that are almost discrete. We also investigate openness of the Salbany lift and a Salbany extension of a continuous function. Related to open continuous functions are initial maps as well as nearly open maps. It turns out that the Salbany map of every space is both initial and nearly open. We repeat the procedure done for openness of Salbany maps, Salbany lifts and Salbany extensions to their initiality and nearly openness. [ABSTRACT FROM AUTHOR]

Let [0,∞) be the set of all non-negative real numbers. The set B[0,∞) = [0,∞) × [0,∞) with the following binary operation (a, b)(c, d) = (a + c - min{b, c}, b + d - min{b, c}) is a bisimple inverse semigroup. In the paper we study Hausdorff locally compact shift-continuous topologies on the semigroup B[0,∞) with an adjoined compact ideal of the following tree types. The semigroup B[0,∞) with the induced usual topology τu from R2, with the topology τL which is generated by the natural partial order on the inverse semigroup B[0,∞), and the discrete topology are denoted by B1 [0,∞), B2 [0,∞), and Bd [0,∞), respectively. We show that if SI 1 (SI 2) is a Hausdorff locally compact semitopological semigroup B1 [0,∞) (B2 [0,∞)) with an adjoined compact ideal I then either I is an open subset of SI 1 (SI 2) or the topological space SI 1 (SI 2) is compact. As a corollary we obtain that the topological space of a Hausdorff locally compact shift-continuous topology on S1 0 = B1 [0,∞) ∪ {0} (resp. S2 0 = B2 [0,∞) ∪ {0}) with an adjoined zero 0 is either homeomorphic to the one-point Alexandroff compactification of the topological space B1 [0,∞) (resp. B2 [0,∞)) or zero is an isolated point of S1 0 (resp. S2 0). Also, we proved that if SId is a Hausdorff locally compact semitopological semigroup Bd [0,∞) with an adjoined compact ideal I then I is an open subset of SId. [ABSTRACT FROM AUTHOR]

We consider Deaconu--Renault groupoids associated to actions of finite-rank free abelian monoids by local homeomorphisms of locally compact Hausdorff spaces. We study simplicity of the twisted C*-algebra of such a groupoid determined by a continuous circle-valued groupoid 2- cocycle. When the groupoid is not minimal, this C*-algebra is never simple, so we focus on minimal groupoids. We describe an action of the quotient of the groupoid by the interior of its isotropy on the spectrum of the twisted C*-algebra of the interior of the isotropy. We prove that the twisted groupoid C*-algebra is simple if and only if this action is minimal. We describe applications to crossed products of topological-graph C*-algebras by quasi-free actions. [ABSTRACT FROM AUTHOR]

Let W be the category of Archimedean lattice-ordered groups with weak order unit, Comp the category of compact Hausdorff spaces, and W Y -! Comp the Yosida functor, which represents a W-object A as consisting of extended real-valued functions A = D(Y A) and uniquely for various features. This yields topological mirrors for various algebraic (W-theoretic) properties providing close analysis of the latter. We apply this to the subclasses of a-projectable, and laterally a-complete objects, denoted P(a) and L(a), where a is a regular infinite cardinal or 8. Each W-object A has unique minimum essential extensions A = p(a)A = l(a)A in the classes P(a) and L(a), respectively, and the spaces Y p(a)A and Y l(a)A are recognizable (for the most part); then we write down what p(a)A and l(a)A are as functions on these spaces. The operators p(a) and l(a) are compared: we show that both preserve closure under all implicit functorial operations which are finitary. The cases of A = C(X) receive special attention. In particular, if (p < a) l(a)C(X) = C(Y l(a)C(X)), then X is finite. But (p = a) for infinite X, p(a)C(X) sometimes is, and sometimes is not, C(Y p(a)C(X)). [ABSTRACT FROM AUTHOR]

In this paper, using the method of Hicks, we prove some new related fixed point theorems for pair of mappings on two d-complete topological spaces under the suitable conditions. We also derive some related fixed point theorems from our main results. Finally, by using w-continuity, we establish a related fixed point theorem in d-complete Hausdorff topological spaces. [ABSTRACT FROM AUTHOR]

Let X be a locally compact Hausdorff topological space, let V=(vn,k)n,k∈N$\mathcal {V}=(v_{n,k})_{n,k\in {\mathbb {N}}}$ be a system of positive continuous functions on X and let φ be a continuous self‐map on X. The composition operators Cφ:f↦f∘φ$C_\varphi : f\mapsto \ f\,\circ\, \varphi$ on the weighted function (LF)‐spaces VC(X)$\mathcal {V}C(X)$ (V0C(X)$\mathcal {V}_0C(X)$, resp.) and on the weighted function (PLB)‐spaces AC(X)$\mathcal {A}C(X)$ (A0C(X)$\mathcal {A}_0C(X)$, resp.) are studied. We characterize when the operator Cφ$C_\varphi$ acts continuously on such spaces in terms of the system V$\mathcal {V}$ and the map φ, as well as we determine conditions on V$\mathcal {V}$ and φ which correspond to various basic properties of the composition operator Cφ$C_\varphi$, like boundedness, compactness, and weak compactness. Our approach requires a study of the continuity, boundedness, (weak) compactness of the linear operators between (LF)‐spaces and (PLB)‐spaces. [ABSTRACT FROM AUTHOR]

Topological spaces with separately continuous Mal'tsev operation, called quasi-Mal'tsev spaces, are considered. The existence of the free quasi-Mal'tsev space generated by an arbitrary completely regular Hausdorff space is proved. It is shown that any quasi-Mal'tsev space is a quotient of a free quasi-Mal'tsev space. It is also shown that the topology of a free quasi-Mal'tsev space has a simple and natural description in terms of the generating space. Finally, it is proved that any completely regular Hausdorff quasi-Mal'tsev space is a retract of a quasi-topological group. [ABSTRACT FROM AUTHOR]

Let {Ax: x ∊ X} be a collection of complex Banach algebras indexed by the compact Hausdorff space X: We investigate the character amenability of certain algebras A of Ax-valued functions in relation to the character amenability of the AX. [ABSTRACT FROM AUTHOR]

By computing the completely bounded norm of the flip map on the Haagerup tensor product C 0 Y 1 ⊗ C 0 X C 0 Y 2 associated to a pair of continuous mappings of locally compact Hausdorff spaces Y 1 → X ← Y 2 , we establish a simple characterization of the Beck-Chevalley condition for base change of operator modules over commutative C ∗ -algebras, and a descent theorem for continuous fields of Hilbert spaces. [ABSTRACT FROM AUTHOR]

A well-known theorem of J. E. Hutchinson states that if an iterated function system consists of similarity transformations and satisfies the open set condition then its attractor supports a self-similar measure with Hausdorff dimension equal to the similarity dimension. In this article we prove the following result which may be regarded as a form of partial converse: if an iterated function system consists of invertible affine transformations whose linear parts do not preserve a common invariant subspace, and its attractor supports a self-affine measure with Hausdorff dimension equal to the affinity dimension, then the system necessarily consists of similarity transformations. We obtain this result by showing that the equilibrium measures of an affine iterated function system are never Bernoulli measures unless the system either is reducible or consists of similarity transformations. The proof builds on earlier results in the thermodynamic formalism of affine iterated function systems due to Bochi, Feng, Käenmäki, Shmerkin and the first named author and also relies on the work of Benoist on the spectral properties of Zariski-dense subsemigroups of reductive linear groups. [ABSTRACT FROM AUTHOR]

Quillen's notion of small object and the Gabriel-Ulmer notion of finitely presentable or generated object are fundamental in homotopy theory and categorical algebra. Do these notions always lead to rather uninteresting classes of objects in categories of topological spaces, such as all finite discrete spaces, or just the empty space, as the examples and remarks in the existing literature may suggest? This article demonstrates that the establishment of full characterizations of these notions (and some natural variations thereof) in many familiar categories of spaces can be quite challenging and may lead to unexpected surprises. In fact, we show that there are significant differences in this regard even amongst the categories defined by the standard separation axioms, with the T1-separation condition standing out. The findings about these specific categories lead us to insights also when considering rather arbitrary full reflective subcategories of the category of all topological spaces. [ABSTRACT FROM AUTHOR]

Let (X, τ) be a topological space, we will denote by |X|,|X|K, |X|τ and |X|δ, the cardinalities of X; the family of compacts in X; the family of closed in X, and the family of Gδ-closed in X, respectively. The purpose of this work is to establish relationships between these four numbers and conditions under which two of them coincide or one of them is ≤ c, where c denotes, as usual, the cardinality of the set of real numbers R. We will use the Stone-Weierstrass theorem to prove that: Let (X, τ) be a compact Hausdorff topological space, then .... [ABSTRACT FROM AUTHOR]

Let A be a unital C⁎-algebra equipped with its natural order. As usual the effect algebra of A is the interval { a ∈ A : 0 ≤ a ≤ I A } , where I A denotes the unit of A. In this paper, we give a complete description of order isomorphisms between effect algebras of C⁎-algebras of type C (X) ⊗ B (H) , where C (X) stands for the algebra of all continuous complex valued functions on a (non pathological) Hausdorff compact space X and B (H) denotes the algebra of all bounded linear operators on a complex Hilbert space H. Our results generalize some works by L. Molnár and P. Šemrl. [ABSTRACT FROM AUTHOR]