Theta Omega Topological Operators and Some Product Theorems

A n-dimensional topological manifold is defined as an n-dimensional local Euclidean space M that is also a second countable space and a Hausdorff space. If a topological space has a countable base, it is referred to as a second countable space. At the same time, continuous function and metric space can help people to understand topological space better. This article will express some basic ideas about continuous functions, metric space, topological space and its properties, and topological manifolds. Moreover, the paper shows some basic ideas of how topological manifolds, topological space and function could be recognized and proved. Through analysis, this paper demonstrates the connection between them, such as using properties of continuous function to prove the definition and properties of topological manifolds and spaces.

In this manuscript, we introduce the idea of operation approach on on spherical fuzzy soft topology (𝒮𝓕𝒮 𝒯) and analyse its characteristics in spherical fuzzy soft topological space. Also, we define 𝒮𝓕𝒮 g~-regular space and 𝒮𝓕𝒮 g~ regular operation in 𝒮𝓕𝒮 𝒯𝒮 and study certain results using 𝒮𝓕𝒮 g~-regular space and 𝒮𝓕𝒮 g~-regular operation. Using operation on 𝒮𝓕𝒮 topology, we bring forth the topological operators like 𝒮𝓕𝒮 g~-closure and 𝒮𝓕𝒮 g~-interior and analyse their characteristics. As an application of operation approach on 𝒮𝓕𝒮 topology, we present a decision making algorithm and provide an illustration. We also develop a python program for the proposed algorithm.

Soft ω-local indiscreetness as a weaker form of both soft local countability and soft local indiscreetness is introduced. Then soft ω-regularity as a weaker form of both soft regularity and soft ω-local indiscreetness is defined and investigated. Additionally, soft ω-T2 as a new soft topological property that lies strictly between soft T2 and soft T1 is defined and investigated. It is proved that soft anti-local countability is a sufficient condition for equivalence between soft ω-locally indiscreetness (resp. soft ω-regularity) and soft locally indiscreetness (resp. soft ω-regularity). Additionally, it is proved that the induced topological spaces of a soft ω-locally indiscrete (resp. soft ω-regular, soft ω-T2) soft topological space are (resp. ω-regular, ω-T2) topological spaces. Additionally, it is proved that the generated soft topological space of a family of ω-locally indiscrete (resp. ω-regular, ω-T2) topological spaces is soft ω-locally indiscrete and vice versa. In addition to these, soft product theorems regarding soft ω-regular and soft ω-T2 soft topological spaces are obtained. Moreover, it is proved that soft ω-regular and soft ω-T2 are hereditarily under soft subspaces.

In this article, we introduce the notion of the complementary soft neighborhood and present three kinds of covering soft rough set (mathcal {CSR}) models. The basic properties of these models are investigated. The relationships among these models are also discussed. Moreover, we establish the topological approach to mathcal {CSR} say, varDelta -topological spaces (i.e., varDelta -TS). Hence, the topological properties for varDelta -TS models such as varDelta -open sets, varDelta -closed sets, varDelta -interior, varDelta -closure, varDelta -boundary, varDelta -neighborhood and varDelta -limit point are studied and the relationships between them are given. Finally, we make use of an algorithm for these proposed models to deal with uncertainties for solving the MGDM problems using the constructed topologies.

As a weaker form of ω-paracompactness, the notion of σ-ω-paracompactness is introduced. Furthermore, as a weaker form of σ-ω-paracompactness, the notion of feebly ω-paracompactness is introduced. It is proven hereinthat locally countable topological spaces are feebly ω-paracompact. Furthermore, it is proven hereinthat countably ω-paracompact σ-ω-paracompact topological spaces are ω-paracompact. Furthermore, it is proven hereinthat σ-ω-paracompactness is inverse invariant under perfect mappings with countable fibers, and as a result, is proven hereinthat ω-paracompactness is inverse invariant under perfect mappings with countable fibers. Furthermore, if A is a locally finite closed covering of a topological space X,τ with each A∈A being ω-paracompact and normal, then X,τ is ω-paracompact and normal, and as a corollary, a sum theorem for ω-paracompact normal topological spaces follows. Moreover, three open questions are raised.

A topological space is called a uqu space [10] if it admits a unique quasi-uniformity. Answering a question [2, Problem B, p. 45] of P. Fletcher and W. F. Lindgren in the affirmative we show in [8] that a topological space X is a uqu space if and only if every interior-preserving open collection of X is finite. (Recall that a collection ℒ of open sets of a topological space is called interior-preserving if the intersection of an arbitrary subcollection of ℒ is open (see e.g. [2, p. 29]).) The main step in the proof of this result in [8] shows that a topological space in which each interior-preserving open collection is finite is a transitive space. (A topological space is called transitive (see e.g. [2, p. 130]) if its fine quasi-uniformity has a base consisting of transitive entourages.) In the first section of this note we prove that each hereditarily compact space is transitive. The result of [8] mentioned above is an immediate consequence of this fact, because, obviously, a topological space in which each interior-preserving open collection is finite is hereditarily compact; see e.g. [2, Theorem 2.36]. Our method of proof also shows that a space is transitive if its fine quasi-uniformity is quasi-pseudo-metrizable. We use this result to prove that the fine quasi-uniformity of a T1 space X is quasi-metrizable if and only if X is a quasi-metrizable space containing only finitely many nonisolated points. This result should be compared with Proposition 2.34 of [2], which says that the fine quasi-uniformity of a regular T1 space has a countable base if and only if it is a metrizable space with only finitely many nonisolated points (see e.g. [11] for related results on uniformities). Another by-product of our investigations is the result that each topological space with a countable network is transitive.

e-2 - Classical Metrization Theorems

In this paper we introduce (∈ − 2)-ball centered at each point in 2-metric topological space (X,d). Theorems on the normal, regular and Hausdorff topological spaces in 2-metrizable topological space are presented. We show that every metrizable topological spaces are coarser than 2-metrizable topological space, and then we conclude that each manifold is a 2-metric topological space.

In this paper notions ofm-Lindelof, meta-m-Lindelof, para-m-Lindelof andm-closure preserving property are defined, wherem is any infinite cardinal. The main results are the following: (a) A topological space ism-Lindelof if and only if it is meta-m-Lindelof and it ism-Lindelof in the sense of complete accumulation point. (b) A regular topological space is paracompact if and only if it is para-m-Lindelof and it hasm-closure preserving property for somem.

‎By substituting the usual notion of open sets in a topological space $X$ with a suitable collection of maps from $X$ to a frame $L$, we introduce the notion of L-topological spaces. Then, we proceed to study the classical notions and properties of usual topological spaces to the newly defined mathematical notion. Our emphasis would be concentrated on the well understood classical connectedness, quotient and compactness notions, where we prove the Thychonoff's theorem and connectedness property for ultra product of $L$-compact and $L$-connected topological spaces, respectively.

The significance fore supra topological spaces as a subject of study cannot be overstated, as they represent a broader framework than traditional topological spaces. Numerous scholars have proposed extension to supra open sets, including supra semi open sets, supra per open and others. In this research, a notion for ⱨ-supra open created within the generalizations of the supra topology of sets. Our investigation involves harnessing this style of sets to introduce modern notions in these spaces, specifically supra ⱨ - interior, supra ⱨ - closure, supra ⱨ - limit points, supra ⱨ - boundary points and supra ⱨ - exterior of sets. It has been examining the relationship with supra open. The research was also enriched with many of characteristics of each concept. Building upon this set classification, we introduced several kinds of maps like supra ⱨ - continuous, supra ⱨ - open, supra ⱨ - tentative, supra ⱨ -globally and supra ⱨ - homeomorphism. Additionally, we have proven a collection of useful relationships for the aforementioned of functions. Furthermore, the research was enhanced with illustrative and refuting examples.

False data injection attack (FDIA) is a typical cyber-attack aiming at falsifying measurement data for state estimation (SE), which may incur catastrophic consequences on cyber-physical system operation. In this paper, we develop a deep learning based methodology for detection, localization, and data recovery of FDIA on power systems in a coherent and holistic manner. However, the multi-modal probability distributions of both measurements and state variables in SE due to ever-changing operating points and structural/topological changes pose great challenges in detecting and localizing FDIA. To address this challenge, we first propose an enhanced attack model to launch massive FDIA on limited access points. Second, we train an auto-encoder (AE) with a Bayesian change verification (BCV) classifier using <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N-1$</tex> contingencies to detect FDIA with unseen <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N-k$</tex> operational topologies. Third, to avoid model collapse caused by multi-modal measurement distribution, an AE-based generative adversarial network (GAN) is derived to generate a diverse candidate set of normal measurement vectors with various operational topologies. Finally, we develop a pattern match algorithm to localize and recover the falsified measurements and state variables by comparing the falsified measurement vectors with the normal measurement vectors in the candidate set. Case studies with IEEE benchmark systems and a modified 415-bus China Southern Grid system are provided to validate the proposed methodology. It shows that the proposed methodology achieves an average 95% accuracy for detection, over 80% accuracy for localization of FDIA, and recovers the measurement and state variables close to their true values.

This paper mainly discuss eight kinds of equivalent characterizations of an M-fuzzifying topological closure operator cl: 2X ? MX, an M-fuzzifying topological interior operator int: 2X ? MX, and an M-fuzzifying topological derived operator der: 2X ? MX, in which M is a completely distributive De Morgan algebra. As its applications, we give the fuzzy constructions of M-fuzzifying topological operators cld, intd, derd induced by an M-fuzzifying pseudo metric d in the sense of Morsi?s fuzzy metric and prove that the continuity between M-fuzzifying topological operator spaces and M-fuzzifying pseudo metric spaces is maintained. Finally, we show that the M-fuzzifying topological T induced by d is exactly the M-fuzzifying topological induced by cld, intd and derd. Namely, Td = Tcld = Tintd = Tderd.

This paper introduces some properties of separation axioms called $alpha$ -feeble regular and $alpha$ -feeble normal spaces (which are weaker than the usual axioms) by using elements of graph which are the essential parts of our $alpha$ -topological spaces that we study them. Also, it presents some dependent concepts and studies their properties and some relationships between them.

We define the ??-closure operator as a new topological operator. We show that ??-closure of a subset of a topological space is strictly between its usual closure and its ?-closure. Moreover, we give several sufficient conditions for the equivalence between ??-closure and usual closure operators, and between ??-closure and ?-closure operators. Also, we use the ??-closure operator to introduce ??-open sets as a new class of sets and we prove that this class of sets lies strictly between the class of open sets and the class of ?-open sets. We investigate ??-open sets, in particular, we obtain a product theorem and several mapping theorems. Moreover, we introduce ?-T2 as a new separation axiom by utilizing ?-open sets, we prove that the class of !-T2 is strictly between the class of T2 topological spaces and the class of T1 topological spaces. We study relationship between ?-T2 and ?-regularity. As main results of this paper, we give a characterization of ?-T2 via ??-closure and we give characterizations of ?-regularity via ??-closure and via ??-open sets.