We define and study new ideas of fibrewise topological space namely fibrewise multi-topological space . We also submit the relevance of fibrewise closed and open topological space . Also fibrewise multi-locally sliceable and fibrewise multi-locally section able multi-topological space . Furthermore, we propose and prove a number of statements about these ideas. On the other hand, extend separation axioms of ordinary topology into fibrewise setting. The separation axioms are said to be fibrewise multi-T0. spaces, fibrewise multi-T1spaces, fibrewise multi-R0 spaces, fibrewise multi-Hausdorff spaces, fibrewise multi-functionally Hausdorff spaces, fibrewise multi-regular spaces, fibrewise multi-completely regular spaces, fibrewise multi-normal

In the present paper we introduce and study new classes of soft separation axioms in soft bitopological spaces, namely, soft (1,2)*-omega separation axioms and weak soft (1,2)*-omega separation axioms by using the concept of soft (1,2)*-omega open sets. The equivalent definitions and basic properties of these types of soft separation axioms also have been studied.

 In this paper, we introduce  and  study  new  classes of soft  open  sets  in soft bitopological spaces called soft  (1,2)*-omega  open  sets  and  weak forms of soft (1,2)*-omega open sets such as soft  (1,2)*-α-ω-open sets, soft  (1,2)*-pre-ω-opensets, soft  (1,2)*-b-ω-open sets,  and  soft  (1,2)*-β-ω-open  sets. Moreover; some basic properties and the relation among these concepts and other concepts also have been studied.

The goal of this article is to construct fibrewise w-compact (resp. locally w-compact) spaces. Some related results and properties of these concepts will be investigated. Furthermore, we investigate various relationships between these concepts and three classes of fibrewise w-separation axioms.

The aim of this paper is to look at fibrewise slightly issuances of the more important separation axioms of ordinary topology namely fibrewise said to be fibrewise slightly T0 spaces, fibrewise slightly T1spaces, fibrewise slightly R0 spaces, fibrewise slightly T2 spaces, fibrewise slightly functionally T2 spaces, fibrewise slightly regular spaces, fibrewise slightly completely regular spaces, fibrewise slightly normal spaces. In addition, we announce and confirm many proposals related to these concepts.

The purpose of this paper is to consider fibrewise near versions of the more important separation axioms of ordinary topology namely fibrewise near T0 spaces, fibrewise near T1 spaces, fibrewise near R0 spaces, fibrewise near Hausdorff spaces, fibrewise near functionally Hausdorff spaces, fibrewise near regular spaces, fibrewise near completely regular spaces, fibrewise near normal spaces and fibrewise near functionally normal spaces. Also we give several results concerning it.

In this paper, we shall introduce a new kind of Perfect (or proper) Mappings, namely ω-Perfect Mappings, which are strictly weaker than perfect mappings. And the following are the main results: (a) Let f : X→Y be ω-perfect mapping of a space X onto a space Y, then X is compact (Lindeloff), if Y is so. (b) Let f : X→Y be ω-perfect mapping of a regular space X onto a space Y. then X is paracompact (strongly paracompact), if Y is so paracompact (strongly paracompact). (c) Let X be a compact space and Y be a p*-space then the projection p : X×Y→Y is a ω-perfect mapping. Hence, X×Y is compact (paracompact, strongly paracompact) if and only if Y is so.

   In this paper, we introduce a new type of functions in bitopological spaces, namely, (1,2)*-proper functions. Also, we study the basic properties and characterizations of these functions . One of the most important of equivalent definitions to the (1,2)*-proper functions is given by using (1,2)*-cluster points of filters . Moreover we define and study (1,2)*-perfect functions and (1,2)*-compact functions in bitopological spaces and we study the relation between (1,2)*-proper functions and each of (1,2)*-closed functions , (1,2)*-perfect functions and (1,2)*-compact functions and we give an example when the converse may not be true .