Fuzzy Retractions of Fuzzy Open Flat Robertson-Walker Space

Our aim in the present paper is to introduce and study new types of fuzzy retractions of fuzzy open flat Robertson-Walker ̃ W4 model. New types of the fuzzy deformation retracts of ̃ W4 model are obtained. The relations between the fuzzy foldings and the fuzzy deformation retracts of ̃ Wmodel are deduced. Types of fuzzyminimal retractions are also presented. New types of homotopy maps are deduced. New types of conditional fuzzy folding are presented. Some commutative diagrams are obtained.


Introduction and Background
Robertson-Walker space represents one of the most intriguing and emblematic discoveries in the history of geometry.Although if it were introduced for a purely geometrical purpose, they came into prominence in many branches of mathematics and physics.This association with applied science and geometry generated synergistic effect: applied science gave relevance to Robertson-Walker space and Robertson-Walker space allowed formalizing practical problems [1][2][3][4][5][6].As is well known, the theory of retractions is always one of interesting topics in Euclidian and Non-Euclidian spaces and it has been investigated from the various viewpoints by many branches of topology and differential geometry [7][8][9][10][11].There are many diverse applications of certain phenomena for which it is impossible to get relevant data.It may not be possible to measure essential parameters of a process such as the temperature inside molten glass or the homogeneity of a mixture inside some tanks.The required measurement scale may not exist at all, such as in the case of evaluation of offensive smells, evaluating the taste of foods or medical diagnoses by touching [7,8,[12][13][14][15][16][17][18].The aim of the present paper is to describe the above phenomena geometrically, specifically concerned with the study of the new types of fuzzy retractions, fuzzy deformation retracts, and fuzzy folding of fuzzy open flat Robertson-Walker W4 model.A fuzzy manifold is manifold which has a physical character.This character is represented by the density function , where  ∈ [0, 1] [7,8,12].
Topological folding of fuzzy open flat Robertson-Walker space W4 model [7,8].A map F : W4 → W4 is said to be an isometric folding of W4 model into itself if and only if for any piecewise fuzzy geodesic path  :  → W4 the induced path F ∘  :  → W4 is a piecewise fuzzy geodesic and of the same length as , where  = [0, 1].If F does not preserve lengths, then F is a topological folding of fuzzy Robertson-Walker space W4 model [12][13][14].
The fuzzy folding of ⋃ M ⊆ W4 model is a folding f : ⋃ M → ⋃ M such that f( M) = M and any M belong to the upper hypermanifolds ∃ M down M such that   =   for every corresponding points, that is, (  ) = (  ) [15].See Figure 1.

Main Results
Theorem 1.The fuzzy retractions of W4 model are the fuzzy unit hyperboloid, fuzzy hyperbolic, fuzzy hypersphere, fuzzy circle, and fuzzy minimal manifolds.

Conclusion
In the present paper, we obtain and study new types of fuzzy retractions of W4 model.Also, we deduced new types of fuzzy deformation retract of W4 model.The relations between the fuzzy folding and the fuzzy deformation retracts of W4 model is obtained.New types of minimal fuzzy retraction of W4 model is also presented.New types of homotopy maps are described.The isometric and topological fuzzy folding in each case and the relation between the fuzzy deformation retract after and before fuzzy folding have been obtained.Types of conditional fuzzy folding of W4 model are described.