Fibrewise Multi-Perfect Topological Spaces

The essential objective of this paper is to introduce new notions of fibrewise topological spaces on D that are named to be upper perfect topological spaces, lower perfect topological spaces, multi-perfect topological spaces, fibrewise upper perfect topological spaces, and fibrewise lower perfect topological spaces. fibrewise multi-perfect topological spaces, filter base, contact point, rigid, multi-rigid, multi-rigid, fibrewise upper weakly closed, fibrewise lower weakly closed, fibrewise multi-weakly closed, set, almost upper perfect, almost lower perfect, almost multi-perfect, fibrewise almost upper perfect, fibrewise almost lower perfect, fibrewise almost multi-perfect, upper * continuous fibrewise upper ∗ topological spaces respectively, lower * continuous fibrewise lower ∗ topological spaces respectively, multi * -continuous fibrewise multi ∗ -topological spaces respectively multi-T e , locally In addition, we find and prove several propositions linked to these notions.


Introduction
We begin our work with the concept of category of Fibrewise (briefly, ..) set on a known set, named the base set.If the base set is stated with D, then a F.W. set on D applied to a set E with a function X is X: E → D, named the projection (briefly, project).For every point d of D, the fiber on d is the subset E d = X −1 (d) of E; fibers will be empty, so we do not require X to be a surjection.Also, for every subset D* of D, we regard E D * = X −1 (D * ) as a ..set on D* with the project determined by X.A multi-function [2] Ω of a set E into F is a correspondence such that Ω (e) is a nonempty subset of F for every e ∈ E. We will denote such a multi-function by Ω: E → F .For a multi-function Ω, the upper and lower inverse set of a set K of F, will be denoted by Ω + (K) and Ω − (K), respectively, that is Ω + (K) = {e ∈ E : Ω(e) ⊆ K} and Ω − (K) = {e ∈ E: Ω(e) ∩ K ≠ ∅ }.Definition 1.1.[7] Suppose that E and F are ..sets on D, with project.  :  →  and   :  → , respectively, a function Ω: E → F is named to be ..if   Ω =   , that is to say if Ω(Xd) ⊂ Fd for every point d of D.
For other concepts or information that are undefined here, we follow nearly [3] and [4] Recall that [7] Let D be a topological space, the ..Topology space (briefly, ..T.S.) on a ..set E on D, which means any topology on E for that the project X is continuous.Remark 1.1.[7] i.The smaller topology is the topology trace with X, where in the open sets of E are the pre image of the open sets of D, this is named the ..indiscrete topology.ii.The ..T.S. on D is stated to be a ..set on D with a ..T.S.
We regard the topology product D × T, for any topological space T, as a ..T.S. on D using the category of the first projection.The equivalences in the category of ..T.S. are named ..T. equivalences.If E is ..T. equivalent to D × T, for some topological space T, we say that E is trivial, as a ..T.S. on D. In ..T. the form neighbourhood (briefly, ℙ) is used in the same sense as it is in normally topology, but the forms ..basic may need some illustration, so let E be ..T.S. on D, if e is a point of Ed where in d ∈ D, appear a family N(e) of ℙ of e in E as ..basic if as every ℙ H of e we have Ew ∩ K ⊂ H, for some element K of N(e) and ℙ W of d in D. As example, in the case of the topological product D × T, where in T is a topological spaces, the family of Cartesian products D × N(t), where in N(t) runs through the ℙ of t, is ..basic for (d, t). 1.2.[7] The ..functions Ω: E→ F; E and F are ..spaces on D is named: Let Ω: E → F be a multi-function.Then Ω is multi cont.(briefly, M. cont.) if it is U. cont.and L. cont.Definition 1.6.[5] Let D be topological space, the ..upper topology space (briefly, ..U.T.S.) on a ..set E on D mean any topology on E for which the project.X is U. cont.

Fibrewise Multi-Perfect and multi-Rigidity Topological Spaces.
In this segment, we present the idea of multi-perfect topological, upper rigidity spaces lower rigidity spaces, multi-rigidity spaces and make sure of some of its base characteristics.A subset  of topological space (E,τ) is named to be multi-rigid in E (briefly, M.R.) if it is U.R. and L.R.

Conclusion
The main purpose of the present work is to providethe starting point for some application of fibr multi-prft tplgical spa structures in a falter base by using multi-topological spaces.Definitions of characterization theorems are used for multi-rgd, fibr multi-akly cld,  t, fibre almt multi-prfct, multi * -ntinuus fibr multi * -tplgical spa.

Definition 1 . 3 . [ 7 ]
(a) Continuous (briefly, cont.) if every e ∈ Ed; d ∈ D, the inverse image of every open set of Ω(e) is an open set of e.(b) Open if for every e∈E_d, d ∈D, the direct image of every open set of e is an open set of Ω(e).The F.W.T.S. E on D is named ..closed (resp., open) if the project.X is closed (resp., open) functions.Definition 1.4.[1]Let Ω: E → F be a multi-function.Then Ω is upper cont.(briefly, U. cont.) if

C𝒐𝒓𝒐llary 4 . 7 .
For a topological space (E,), the next are equivalent: i. H s QHC.ii.A . .M. (E,) is P.T space th constant projection on D * in which D * is a singleton with two equal topologies meaning the unique topology on D * .iii.The . .. (B×H,Q) is M.P.T.S. on (D,), in which  =  × .

Definition 1.7.[5]
Let D be topological space the ..lower topology space (briefly, ..L.T.S.) on a ..set E on D mean any topology on E for which the project.X is L. cont.Let D be topological space the ..multi-topology space (briefly, ..M.T.S.) if it is ..U.T.S. and ..L.T.S.A filtr ℑ on topological space (E,τ) a non-empty collection of non-empty subsets of E such that i. ∀ 1,  2 ∈ ℑ,  1 ∩  2 ∈ ℑ ii.If  1 ⊆  2 ⊆E and 1∈ ℑ then 2 ∈ ℑ.If ℑ, filter bases on (E,), we namely  is finr than ℑ (writtn as ℑ < ) if for all  ∈ ℑ, there is G ⊆  meets if ∩G ≠ ∅ for vry  ∈ ℑ and G ∈ .Definition 1.10.[10] If E is topological space and e ∈ E a ℙ of e is a st  which contain an opn st V containing e.If  is opn st and contains e w namly  is opn  ℙ for a point e.

Definition 1.11. [9] A
pint e in (E,) is namd t b a contact point of a subst  ⊆ E ff ∀  open ℙ of e, cl () ∩  ≠ ∅.So, st of all cntact pints of  is namd t b th closure of  and is symbolizd by cl ().

Definition 1.13. [2] L𝑒t
e a point in a ..T.S. (E,) on (D,) is namd to b adhrnt point f a F * .B * .ℑ. on E (brifly, ad(e)) ff all number of ℑ is contract a point.A set of all adherent point of ℑ is namd to b th adhrnc of ℑ and is symbolizs by ad(ℑ).