We use the technique of Hattori to generate new topologies on the closed upper half plane which lie between the usual metric topology and the Niemytzki topology. We study some of their fundamental properties and weaker versions of normality.

The concepts of soft faint continuity as a weaker form of soft weak continuity and soft faint ${\theta }_{\omega }$-continuity as a weaker form of soft weak ${\theta }_{\omega }$-continuity are introduced. Numerous characterizations of them are given. We further demonstrate that, under soft restrictions, they are retained. Moreover, we show that a soft function is soft faintly continuous (respectively, soft faintly ${\theta }_{\omega }$-continuous) if its soft graph function is soft faintly continuous (respectively, soft faintly ${\theta }_{\omega }$-continuous). In addition, we show that a soft function with a soft almost regular (respectively, soft extremally disconnected) co-domain is soft faintly continuous iff it is soft almost continuous (respectively, soft $\delta$-continuous). Furthermore, we show that soft faintly continuous surjective functions are soft set-connected functions, and as a corollary, we demonstrate how soft faintly continuous functions sustain soft connectivity. Finally, we studied the symmetry between our new notions and their topological counterparts.

As an extension of quasi-continuity in general topology, we define soft quasi-continuity. We show that this notion is equivalent to the known notion of soft semi-continuity. Next, we define soft weak quasi-continuity. With the help of examples, we prove that soft weak quasi-continuity is strictly weaker than both soft semi-continuity and soft weak continuity. We introduce many characterizations of soft weak quasi-continuity. Moreover, we study the relationship between soft quasi-continuity and weak quasi-continuity with their analogous notions in general topology. Furthermore, we show that soft regularity of the co-domain of a soft function is a sufficient condition for equivalence between soft semi-continuity and soft weakly quasi-continuity. Furthermore, we provide several results of soft composition, restrictions, preservation, and soft graph theorems in terms of soft weak quasi-continuity.

In this paper, we present a novel family of soft sets named “soft ${\omega }_{\delta }$-open sets”. We find that this class constitutes a soft topology that lies strictly between the soft topologies of soft $\delta$-open sets and soft ${\omega }^0$-open sets. Also, we introduce certain sufficient conditions for the equivalence between this new soft topology and several existing soft topologies. Moreover, we verify several relationships that contain soft covering properties, such as soft compactness and soft Lindelofness, which are related to this new soft topology. Furthermore, in terms of the soft interior operator in certain soft topologies, we define four classes of soft sets. Via them, we obtain new decomposition theorems for soft $\delta$-openness and soft $\theta$-openness, and we characterize the soft topological spaces that have the soft “semi-regularization property”. In addition, via soft ${\omega }_{\delta }$-open sets, we introduce and investigate a new class of soft functions named “soft ${\omega }_{\delta }$-continuous functions”. Finally, we look into the connections between the newly proposed soft concepts and their counterparts in classical topological spaces.

Let (X, tau) be a topological space and p is not an element of X. Put X-p = X boolean OR { p }. Define a topology tau* on X-p by tau* ={phi}boolean OR{U boolean OR{p}:U is an element of tau}. The space (X-p, tau*) is called the closed extension space of (X, tau). We present new results about the closed extension topological spaces. Mainly weaker versions of normality.

Presently, the scientists across the world are carrying out the theoretical as well as the experimental examinations for describing the importance of nanofluid in the heat transfer phenomena. Such fluids can be obtained by suspending nanoparticles in base fluid. Experimentally, it has proved that the thermal characteristics of nanofluid are much better and appealing as compared to traditional fluid. The current work investigates the heat transfer for flow of blood that comprises of micropolar gold nanoparticles. A microorganism creation also affects the concentration of nanoparticles inside the channel. Suitable transformation has used to change the mathematical model to dimensionless form and then have solved by employing the homotopy analysis method. In this investigation it has revealed that, fluid?s motion decays with growth in Reynolds, Darcy numbers and volumetric fraction. Thermal characteristics support by augmentation in volumetric fraction, while oppose by Prandtl number. Density of microorganism weakens by growth in Peclet and bioconvection Lewis numbers.