The theory of Topological Space Fiber is a new and essential branch of mathematics, less than three decades old, which is created in forced topologies. It was a very useful tool and played a central role in the theory of symmetry. Furthermore, interdependence is one of the main things considered in topology fiber theory. In this regard, we present the concept of topological spaces α associated with them and study the most important results.

Form the series of generalization of the topic of supra topology is the generalization of separation axioms . In this paper we have been introduced (S * - SS *) regular spaces . Most of the properties of both spaces have been investigated and reinforced with examples . In the last part we presented the notations of supra *- -space ( =0,1) and we studied their relationship with (S * - SS *) regular spaces.

Abstract. In this study, we shall research the fibrewise micro ideal topological spaces over Ḃ, as well as the relationship between fibrewise micro ideal topological spaces over Ḃ and fibrewise micro topological spaces over Ḃ. At first present introduces a novel notion from fibrewise micro ideal topological spaces over Ḃ, and differentiates it from fibrewise micro topological spaces over Ḃ. Some fundamental characteristics from these spaces are studied. Then show discussed the fibrewise micro ideal closed and micro ideal open topologies. Many propositions relating to these ideas are offered. In the next part will study defines and investigates novel conceptions from fibrewise micro ideal topological spaces over Ḃ, particularly f

The main purpose of this paper is to introduce a some concepts in fibrewise bitopological spaces which are called fibrewise , fibrewise -closed, fibrewise −compact, fibrewise -perfect, fibrewise weakly -closed, fibrewise almost -perfect, fibrewise ∗-bitopological space respectively. In addition the concepts as - contact point, ij-adherent point, filter, filter base, ij-converges to a subset, ij-directed toward a set, -continuous, -closed functions, -rigid set, -continuous functions, weakly ijclosed, ij-H-set, almost ij-perfect, ∗-continuous, pairwise Urysohn space, locally ij-QHC bitopological space are introduced and the main concept in this paper is fibrewise -perfect bitopological spaces. Several theorems and characterizations c

The aim of this research is to study some types of fibrewise fuzzy topological spaces. The six major goals are explored in this thesis. The very first goal, introduce and study the notions types of fibrewise topological spaces, namely fibrewise fuzzy j-topological spaces, Also, we introduce the concepts of fibrewise j-closed fuzzy topological spaces, fibrewise j-open fuzzy topological spaces, fibrewise locally sliceable fuzzy j-topological spaces and fibrewise locally sectionable fuzzy j-topological spaces. Furthermore, we state and prove several Theorems concerning these concepts, where j={δ,θ,α,p,s,b,β} The second goal is to introduce weak and strong forms of fibrewise fuzzy ω-topological spaces, namely the fibrewise fuz

Most real-life situations need some sort of approximation to fit mathematical models. The beauty of using topology in approximation is achieved via obtaining approximation for qualitative subgraphs without coding or using assumption. The aim of this paper is to apply near concepts in the -closure approximation spaces. The basic notions of near approximations are introduced and sufficiently illustrated. Near approximations are considered as mathematical tools to modify the approximations of graphs. Moreover, proved results, examples, and counterexamples are provided.

The primary objective of this paper is to present a new concept of fibrewise topological spaces over B is said to be fibrewise slightly topological spaces over B. Also, we introduce the concepts of fibrewise slightly perfect topological spaces, filter base, contact point, slightly convergent, slightly directed toward a set, slightly adherent point, slightly rigid, fibrewise slightly weakly closed, H.set, fibrewise almost slightly perfect, slightly∗ .continuous fibrewise slightly∗ topological spaces respectively, slightly Te, locally QHC, In addition, we state and prove several propositions related to these concepts.

In this paper, certain types of regularity of topological spaces have been highlighted, which fall within the study of generalizations of separation axioms. One of the important axioms of separation is what is called regularity, and the spaces that have this property are not few, and the most important of these spaces are Euclidean spaces. Therefore, limiting this important concept to topology is within a narrow framework, which necessitates the use of generalized open sets to obtain more good characteristics and preserve the properties achieved in general topology. Perhaps the reader will realize through the research that our generalization preserved most of the characteristics, the most important of which is the hereditary property. Two t

Spelling correction is considered a challenging task for resource-scarce languages. The Arabic language is one of these resource-scarce languages, which suffers from the absence of a large spelling correction dataset, thus datasets injected with artificial errors are used to overcome this problem. In this paper, we trained the Text-to-Text Transfer Transformer (T5) model using artificial errors to correct Arabic soft spelling mistakes. Our T5 model can correct 97.8% of the artificial errors that were injected into the test set. Additionally, our T5 model achieves a character error rate (CER) of 0.77% on a set that contains real soft spelling mistakes. We achieved these results using a 4-layer T5 model trained with a 90% error inject