The product  of   rn-paracompact and   rn-strongly  paracompact are briefly disc. ussed.

In this article an attempt has been made to procure the concept of pairwise neutrosophic simply open set, pairwise neutrosophic simply continuous mapping, pairwise neutrosophic simply open mapping, pairwise neutrosophic simply compactness, pairwise neutrosophic simply b-open set, pairwise neutrosophic simply b-continuous mapping, pairwise neutrosophic simply b-open mapping and pairwise neutrosophic simply b-compactness via neutrosophic bi-topological spaces (in short NBTS). Besides, we furnish few illustrative examples on them via NBTS. Further, we investigate some basic properties of them, and formulate several results on NBTSs.

A complete metric space is a well-known concept. Kreyszig shows that every non-complete metric space  can be developed into a complete metric space , referred to as completion of .

We use the b-Cauchy sequence to form  which “is the set of all b-Cauchy sequences equivalence classes”. After that, we prove  to be a 2-normed space. Then, we construct an isometric by defining the function from  to ; thus  and  are isometric, where  is the subset of  composed of the equivalence classes that contains constant b-Cauchy sequences. Finally, we prove that  is dense in ,  is complete and the uniqueness of  is up to isometrics

The aim of this paper is to introduce and study the notion type 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}.

Since his first existence on earth, human had formed a connecting link for a regressive, kinetic and developed relationship that comes from a semi-complicated interaction between natural environment and constructed environment, and this resulted in the survival of human and his existence continuance. Constructed environment enabled human to survive the natural environment inconstancies and enemies as predators, also it helped him to feel safe, comfortable and to practice his everyday life activities...etc. This alternative interaction resulted in creating a civilized legacy for a group of landmarks that tell about the development of this relationship by elemental output that reached us either by documents and manuscripts or as an existed

      In this paper, a modified three-step iteration algorithm for approximating a joint fixed point of non-expansive and contraction mapping is studied. Under appropriate conditions, several strong convergence theorems and Δ-convergence theorems are established in a complete CAT (0) space. a numerical example is introduced to show that this modified iteration algorithm is faster than other iteration algorithms. Finally, we prove that the modified iteration algorithm is stable. Therefore these results are extended and improved to a novel results that are stated by other researchers. Our results are also complement to many well-known theorems in the literature. This type of research can be played a vital role in computer programming

The phenotypic characteristics in the interior spaces are seeing the result of the ability of the designer in his handling of the vocabulary and the elements to deliver a specific meaning for the recipient , and is working to stir up the receiver and make it effective in the process of perception of space. So the theme of the role of phenotypic characteristics is of great significance in the process of analyzing spaces to reach the goal of the main idea , and show those qualities through relationships design in terms of shape, color and texture ... etc. , to reach also designs more beautiful , and creating an internal environment , creative and continuous with its external environment , Hence the importance of research in that it tries t

This paper consist some new generalizations of some definitions such: j-ω-closure converge to a point,  j-ω-closure directed toward a set, almost  j-ω-converges to a set, almost  j-ω-cluster point, a set  j-ω-H-closed relative, j-ω-closure continuous mappings, j-ω-weakly continuous mappings, j-ω-compact mappings, j-ω-rigid a set, almost j-ω-closed mappings and  j-ω-perfect mappings. Also, we prove several results concerning it, where j ÃŽ{q, δ,a, pre, b, b}.

The main purpose of this paper is to investigate some results. When h is  -( ,δ) – Derivation on prime Γ-near-ring G and K is a nonzero semi-group ideal of G, then G is commutative .