Related papers: Aura Topological Spaces and Generalized Open Sets …

We introduce the concept of a soft aura topological space $(X, \tilde{\tau}, \mathfrak{a}_E)$, obtained by equipping a soft topological space $(X, \tilde{\tau}, E)$ with a soft scope function $\mathfrak{a}_E : X \to \tilde{\tau}$ satisfying…

We combine an ideal topological space $(X, \tau, \mathcal{I})$ with a scope function $\mathfrak{a}: X \to \tau$, $x \in \mathfrak{a}(x)$, to form what we call an ideal-aura topological space $(X, \tau, \mathcal{I}, \mathfrak{a})$. The…

This is the second paper in a series on aura topological spaces $(X, \tau, \mathfrak{a})$, where $\mathfrak{a}: X \to \tau$ is a scope function with $x \in \mathfrak{a}(x)$. We study covering and connectivity properties in this setting.…

Generalized topological spaces in the sense of Cs\'{a}sz\'{a}r have two main features which distinguish them from typical topologies. First, these families of subsets are not closed under intersections. Second, we allow for the possibility…

A Cech closure space $(X,u)$ is a set $X$ with a (Cech) closure operator $u$ which need not be idempotent. Many properties which hold in topological spaces hold in Cech closure spaces as well. The notions of proper (splitting) and…

A celebrated 1922 theorem of Kuratowski states that there are at most 14 distinct sets arising from applying the operations of complementation and closure, any number of times, in any order, to a subset of a topological space. In this paper…

We initiate the study of sheaves on Cech closure spaces, providing a new, unified approach to sheaf theory on many of the major classes of spaces of interest to applications: topological spaces, finite simplicial complexes (seen as $T_0$…

We set up, purely in A-model terms, a novel formalism for the global solution of the open and closed topological A-model on toric Calabi-Yau threefolds. The starting point is to build on recent progress in the mathematical theory of open…

Theories of rough sets and soft sets are powerful mathematical tools for modelling various types of vagueness. Hybrid model combining a rough set with a soft set which is called soft rough set proposed by Feng et al. [3] in 2010. In this…

A famous theorem of Kuratowski states that in a topological space, at most 14 distinct sets can be produced by repeatedly applying the operations of closure and complement to a given set. We re-examine this theorem in the setting of formal…

Leveraging the infinite dimensional neural network architecture we proposed in arXiv:2109.13512v4 and which can process inputs from Fr\'echet spaces, and using the universal approximation property shown therein, we now largely extend the…

The paper is devoted to the study of topologies on the group Aut(X,B) of all Borel automorphisms of a standard Borel space $(X, B)$. Several topologies are introduced and all possible relations between them are found. One of these…

An open (resp., closed) subset A of a topological space (X, T ) is called C-open (resp., C-closed) set if cl(A) \ A (resp., A \ int(A)) is a countable set. This paper aims to present the concept of C-open and C-closed sets. We first…

The main purpose of this paper is to introduce and study two new operators $(\cdot)_R^{\diamond}$ and $cl_R^{\diamond}(\cdot)$ via primal which is a new notion. We also show that the operator $cl_R^{\diamond}(\cdot)$ is a Kuratowski closure…

Covering-based rough set theory is a useful tool to deal with inexact, uncertain or vague knowledge in information systems. Topology, one of the most important subjects in mathematics, provides mathematical tools and interesting topics in…

A topology on a set $X$ is the same as a projection (i.e. an idempotent linear operator) $cl:2^X\to 2^X$ satisfying $A\subset cl(A)$ for all $A\subset X$. That's a good way to summarize Kuratowski's closure operator. Basic geometry on a set…

We pose the following new variant of the Kuratowski closure-complement problem: How many distinct sets may be obtained by starting with a set $A$ of a Polish space $X$, and applying only closure, complementation, and the $d$ operator, as…

A two-dimensional topological sigma-model on a generalized Calabi-Yau target space $X$ is defined. The model is constructed in Batalin-Vilkovisky formalism using only a generalized complex structure $J$ and a pure spinor $\rho$ on $X$. In…

Topologies can be expanded with the help of ideals, using the local function, an operator resembling the closure of a set. The aim of this paper is to define the ideals which enable us to create this topology $\tau^{*}$ on $X$…

This paper covers some recent progress in the study of sg-open sets, sg-compact spaces, N-scattered spaces and some related concepts. A subset $A$ of a topological space $(X,\tau)$ is called sg-closed if the semi-closure of $A$ is included…