CyberSec Research

Let $\ell^{p}$, $1\leq p<\infty$, be the Banach space of absolutely $p$-th power summable sequences and let $\pi_{n}$ be the natural projection to the $n$-th coordinate for $n\in\mathbb{N}$. Let $\mathfrak{W}=\{w_{n}\}_{n=1}^{\infty}$ be a bounded sequence of complex numbers. Define the operator $D_{\mathfrak{W}}: \ell^{p}\rightarrow\ell^{p}$ by, for any $x=(x_{1},x_{2},\ldots)\in \ell^p$, $\pi_{n}\circ D_{\mathfrak{W}}(x)=w_{n}x_{n}$ for all $n\geq1$. We call $D_{\mathfrak{W}}$ a diagonal operator on $\ell^{p}$. In this article, we study the topological conjugate classification of the diagonal operators on $\ell^{p}$. More precisely, we obtained the following results. $D_{\mathfrak{W}}$ and $D_{\vert\mathfrak{W}\vert}$ are topologically conjugate, where $\vert\mathfrak{W}\vert=\{\vert w_{n}\vert\}_{n=1}^{\infty}$. If $\inf_{n}\vert w_n\vert>1$, then $D_{\mathfrak{W}}$ is topologically conjugate to $2\mathbf{I}$, where $\mathbf{I}$ means the identity operator. Similarly, if $\inf_{n}\vert w_n\vert>0$ and $\sup_{n}\vert w_n\vert<1$, then $D_{\mathfrak{W}}$ is topologically conjugate to $\frac{1}{2}\mathbf{I}$. In addition, if $\inf_{n}\vert w_n\vert=1$ and $\inf_{n}\vert t_n\vert>1$, then $D_{\mathfrak{W}}$ and $D_{\mathfrak{T}}$ are not topologically conjugate.

Uniformly star superparacompactness, which is a topological property between compactness and completeness, can be characterized using finite-component covers and a measure of strong local compactness. Using these finite-component covers and the associated functional, we introduce and investigate a variational notion of uniformly star superparacompact subsets in metric spaces in the spirit of studies on uniformly paracompact subset and UC-subset. We show that the collection of all such subsets forms a bornology with a closed base, which is contained in the bornology of uniformly paracompact subsets. Conditions under which these two bornologies coincide are specified. Furthermore, we provide several new characterizations of uniformly star superparacompact metric spaces also known as cofinally Bourbaki-quasi complete spaces in terms of some geometric functionals. As a consequence, we establish new relationships among metric spaces that lie between compactness and completeness.

We consider a special type of self-similar sets, called fractal squares, and give a brief review on recent results and unsolved issues with an emphasis on their topological properties.

According to Cartan, given an ideal $\mathcal I$ of $\mathbb N$, a sequence $(x_n)_{n\in\mathbb N}$ in the circle group $\mathbb T$ is said to {\em $\mathcal I$-converge} to a point $x\in \mathbb T$ if $\{n\in \mathbb N: x_n \not \in U\}\in \mathcal I$ for every neighborhood $U$ of $x$ in $\mathbb T$. For a sequence $\mathbf u=(u_n)_{n\in\mathbb N}$ in $\mathbb Z$, let $$t_{\mathbf u}^\mathcal I(\mathbb T) :=\{x\in \mathbb T: u_nx \ \text{$\mathcal I$-converges to}\ 0 \}.$$ This set is a Borel (hence, Polishable) subgroup of $\mathbb T$ with many nice properties, largely studied in the case when $\mathcal I = \mathcal F in$ is the ideal of all finite subsets of $\mathbb N$ (so $\mathcal F in$-convergence coincides with the usual one) for its remarkable connection to topological algebra, descriptive set theory and harmonic analysis. We give a complete element-wise description of $t_{\mathbf u}^\mathcal I(\mathbb T)$ when $u_n\mid u_{n+1}$ for every $n\in\mathbb N$ and under suitable hypotheses on $\mathcal I$. In the special case when $\mathcal I =\mathcal F in$, we obtain an alternative proof of a simplified version of a known result.

Based on the intuitive notion of convexity, we formulate a universal property defining interval objects in a category with finite products. Interval objects are structures corresponding to closed intervals of the real line, but their definition does not assume a pre-existing notion of real number. The universal property characterises such structures up to isomorphism, supports the definition of functions between intervals, and provides a means of verifying identities between functions. In the category of sets, the universal property characterises closed intervals of real numbers with nonempty interior. In the the category of topological spaces, we obtain intervals with the Euclidean topology. We also prove that every elementary topos with natural numbers object contains an interval object; furthermore, we characterise interval objects as intervals of real numbers in the Cauchy completion of the rational numbers within the Dedekind reals.

This study defines an orbitwise expansive point (OE) as a point, such as $x$ in a metric space $(X,\rho)$, if there is a number $d>0$ such that the orbits of a few points inside an arbitrary open sphere will maintain a distance greater than $d$ from the corresponding points of the orbit of $x$ at least once. The point $x$ is referred to as the relatively orbitwise expansive point (ROE) in the previously described scenario if $d$ is replaced with the radius of the open sphere whose orbit is investigated and whose centre is $x$. %The function generating the orbit is considered to be continuous. We also define OE (ROE) set. We prove that arbitrary union of OE (ROE) set is again OE (ROE) set and every limit point of an OE set is an OE point. We show that, rather than the other way around, Utz's expansive map or Kato's CW-expansive map implies OE (ROE) map. We utilise the concept of OE(ROE) to analyse a time-varying dynamical system and investigate its relevance to certain traits associated with expansiveness.

We introduce fibrewise compactifications in both the setting of locally compact Hausdorff spaces and continuous maps, and the parallel setting of $C^*$-algebras and nondegenerate multiplier-valued $*$-homomorphisms. In both situations, we use fibrewise compactifications to define regulated limits. In the topological setting, regulated limits extend classical inverse limits so that the resulting limit space remains locally compact; examples include the path spaces of directed graphs. In the operator-algebraic setting, regulated limits realise a direct-limit construction for multiplier-valued $*$-homomorphisms; examples include the cores of relative Cuntz-Pimsner algebras.

We intend to study the uniqueness of the Hahn-Banach extensions of linear functionals on a subspace in locally convex spaces. Various characterizations are derived when a subspace $Y$ has an analogous version of property-U (introduced by Phelps) in a locally convex space, referred to as the property-SNP. We characterize spaces where every subspace has this property. It is demonstrated that a subspace $M$ of a Banach space $E$ has property-U if and only if the subspace $M$ of the locally convex space $E$ endowed with the weak topology has the property-SNP, mentioned above. This investigation circles around exploring the potential connections between the family of seminorms and the unique extension of functionals previously mentioned. We extensively studied this property on the spaces of continuous functions on Tychonoff spaces endowed with the topology of pointwise convergence.

In 1961, Palais showed that every smooth proper Lie group action on a smooth manifold admits a compatible Riemannian metric on the manifold such that the action becomes isometric. In 2006, Yoshino studied a continuous proper action of a locally compact Hausdorff group on a locally compact Hausdorff space, and showed that the space carries a compatible uniform structure making the action equi continuous in an appropriate setting. In this paper, we focus on bornological proper actions on bornological spaces and prove that the space admits a compatible coarse structure such that the action becomes equi controlled.

The paper contains two natural constructions of extreme hyperspace selections generated by special ordinal decompositions of the underlying space. These constructions are very efficient not only in simplifying arguments but also in clarifying the ideas behind several known results. They are also crucial in obtaining some new results for such extreme selections. This is achieved by using special sets called clopen modulo a point. Such sets are naturally generated by a relation between closed sets and points of the space with respect to a given hyperspace selection.

This paper investigates spaces equipped with a family of metric-like functions satisfying certain axioms. These functions provide a unified framework for defining topology, uniformity, and diffeology. The framework is based on a family of metric-like functions originally introduced for spaces of submanifolds. We show that the topologies, uniformities, and diffeologies of these spaces can be systematically derived from the proposed axioms. Furthermore, the framework covers examples such as spaces with compact-open topologies, tiling spaces, and spaces of graphs, which have appeared in different contexts. These results support the study of spaces with metric-like structures from both topological and diffeological perspectives.

This paper addresses the topological structures induced on vector spaces by convex modulars that do not satisfy the $\Delta_2$ condition, with particular focus on their applications to variable exponent spaces such as \( \ell^{(p_n)} \) and \( L^{p(\cdot)} \). The motivation behind this investigation is its applicability to the study of boundary value problems involving the variable exponent $p(x)$-Laplacian when $p(x)$ is unbounded, a line of research recently opened by the authors. Fundamental topological properties are analyzed, including separation axioms, countability axioms, and the relationship between modular convergence and classical topological concepts such as continuity. Attention is given to the relation between modular and norm topologies. Special emphasis is placed on the openness of modular balls, the impact of the \(\Delta_2\)-condition, and duality with respect to modular topologies.

We continue the development of the theory of capturing schemes over $\omega_1$ by analyzing the relation between the capturing construction schemes (whose existence is implied by Jensen's $\Diamond$-principle) and both the Continuum Hypothesis and Ostaszewski's $\clubsuit$-principle. Formally, we show that the property of being capturing can be viewed as the conjunction of two properties, one of which is implied by $\clubsuit$ and the other one by CH. We apply these principles to construct multiple gaps, entangled sets and metric spaces without uncountable monotone subspaces.

Recent results of Ciraulo are used to prove that operators $c$, $i$ on an arbitrary poset subject to the usual general closure and interior axioms -- but not subject to the usual duality, for there is no complement -- always generate one of $18$ different monoids under composition. We also show that there are no missing edges in a certain Hasse diagram conjectured by Ciraulo to represent the interior-pseudocomplement monoid on an arbitrary poset.

We introduce and study a new type of mappings in metric spaces termed $n$-point Kannan-type mappings. A fixed-point theorem is proved for these mappings. In general case such mappings are discontinuous in the domain but necessarily continuous at fixed points. Conditions under which usual Kannan mappings and mapping contracting the total pairwise distances between $n$ points are $n$-point Kannan-type mappings are found. It is shown that additional conditions of asymptotic regularity and continuity allow to extend the value of the contraction coefficient in fixed-point theorems for $n$-point Kannan-type mappings.

The comonotonic maxitivity property of functionals frequently appears in the characterization of fuzzy integrals based on the maximum operation. In some special cases, comonotonic maxitivity implies monotonicity of functionals. The question of whether this implication holds in general was posed by T. Radul (2023). It was shown in that paper that the implication is valid for finite compacta. In this article, we provide a negative answer to the general problem and discuss additional properties that need to be imposed to ensure the implication holds.

This paper investigates the interplay between properties of a topological space $X$, in particular of its natural order, and properties of the lax comma category $\mathsf{Top} \Downarrow X$, where $\mathsf{Top}$ denotes the category of topologicalspaces and continuous maps. Namely, it is shown that, whenever $X$ is a topological $\bigwedge$-semilattice, the canonical forgetful functor $\mathsf{Top} \Downarrow X \to \mathsf{Top}$ is topological, preserves and reflects exponentials, and preserves effective descent morphisms. Moreover, under additional conditions on $X$, a characterisation of effective descent morphisms is obtained.

In this thesis, we introduce the subject of D-spaces and some of its most important open problems which are related to well known covering properties. We then introduce a new approach for studying D-spaces and covering properties in general. We start by defining a topology on the family of all principal ultrafilters of a set $X$ called the principal ultrafilter topology. We show that each open neighborhood assignment could be transformed uniquely to a special continuous map using the principal ultrafilter topology. We study some structures related to this special continuous map in the category Top, then we obtain a characterization of D-spaces via this map. Finally, we prove some results on Lindel\"of, paracompact, and metacompact spaces that are related to the property D.