Fuzzy slicing structure property with some results

Introduction

The fundamental distinction from modern conceptions of fuzzy fiber spaces. Seifert believed to be now known as the base space (fuzzy topological space) of fuzzy fiber (topological) space 𝜇 𝑌 was a quotient space of 𝜇 𝑌 and not a component of the structure. Hassler Whitney first described fuzzy fiber space [3] under the term sphere space. Later, he modified the name to sphere bundle [14] . J Mahalakshmi, M Sudha [9] , and others are credited with the theory of fuzzy fiber spaces, fuzzy principal bundles, fuzzy vector bundles, and fuzzy fibrations as special cases. A fuzzy fiber structure (𝜇 𝑋 , 𝜋, 𝜇 𝑌) is a triple consisting of two fuzzy topological spaces 𝜇 𝑋 , 𝜇 𝑌 and 𝜋: 𝜇 𝑋 → 𝜇 𝑌 a continuous surjection. The total fuzzy (or fibered) space refer to space 𝜇 𝑋 , 𝜋is ter med the projection, and 𝜇 𝑌 is the base space for each 𝑦 𝑜 ∈ 𝜇 𝑌 the set 𝐹 = 𝜋 -1 (𝑦 𝑜 ) and 𝐹 is called fuzzy fiber over 𝑦 𝑜 . The fuuzy fiber structure over 𝜇 𝑌 refers to (𝜇 𝑋 , 𝜋, 𝜇 𝑌 ) [9] . A fuzzy fibration is a continuous mapping the fuzzy satisfying lifting homotopy property, and its CW-complex is called fuzzy Serre fibration. The slicing structure property (SSP) introduced by [4] if we have given map 𝑝: 𝐸 → 𝑋 has the SSP then 𝐸 is called sliced fiber space over 𝑋 relative to 𝑝 and in particular, every bundle space is a sliced fiber space. In addition, the triple (𝐸, 𝑝, 𝑋) is called a 𝐻𝑢 fiber space (for more details see) [5] . We investigate that most of the theorems that are valid for fuzzy fibration (fuzzy Serre fibration) [4,6] . In particular, with fuzzy slicing structure property.

Preliminaries

In this section, we illustrate some basic definitions, properties, and results. Definition 2.1 [12] Let 𝑆 and 𝑇 be topological spaces. If 𝑝 and 𝑞 are mapping of 𝑆 into 𝑇, then 𝑝 and 𝑞 are homotopic (𝑝 ≃ 𝑞) if there exists a mapping ℎ: 𝑆 × 𝐼 1 → 𝑇 via ℎ(𝑒, 0) = 𝑝(𝑒) and ℎ(𝑒, 1) = 𝑞(𝑒) for all 𝑒 ∈ 𝑆. The mapping ℎ is called a homotopy between 𝑝 and 𝑞. Definition 2.2 [8,2] Let 𝑝: 𝐸 → 𝐵 be a map (fiber bundles) from topological space 𝐸 to topological space 𝐵, we say that 𝑝 has lifting Homotopy Property (L.H.P.by short) with respect to 𝑋 iff given a map 𝑓: 𝑋 → 𝐸 and ℎ 𝑡: 𝑋 → 𝐵 is homotopy where 𝑝 ∘ 𝑓 = ℎ 𝑜 . Then there exist a homotopy ℎ 𝑡 *: 𝑋 → 𝐸 such that (1)ℎ 𝑜 𝑡 = 𝑓. (2) 𝑝 ∘ ℎ 𝑡 * = ℎ 𝑡 for all 𝑥 ∈ 𝑋 𝑎𝑛𝑑 𝑡 ∈ 𝐼. Where 𝐼 is the unit interval. 1. The map 𝑝 is called Hurewics (Serre) fibration if it has lifting homotopy property with respect to all (CW-complex 𝑌) spaces. 2. The map 𝑝 is fiber space which called fibration if it has lifting homotopy property with respect to classes of space. Definition 2.3 [10] A fuzzy set in set 𝑋 is ordered pair (𝑋, 𝜇 𝐴 (𝑋)) such that 𝜇 𝐴 (𝑋) called membership with domain 𝑋 and the co-domain is value in 𝐼 = [0,1], denoted by = {(𝑥. 𝜇 𝐴 (𝑥) ∶ 𝑥 ∈ 𝑋} Definition 2.4 [3] Let 𝜇 𝐴 ∈ ℌ(𝐸) all the fuzzy set from topological space (𝐸, 𝜏) and 𝜏 * ∈ ℌ(𝐸). Definition 2.6 Let (µ 𝐸 , 𝜋, µ 𝐵 ) be fuzzy fiber space where fuzzy section 𝑠 is a fuzzy continuous 𝑠: µ 𝐵 → µ 𝐸 such that 𝜋 ∘ 𝑠 = 𝑖𝑑 µ 𝐵 . Definition 2.7 A fuzzy fiber structure is a triple (𝜇 𝑋 , 𝜋, 𝜇 𝑌 ) consisting of two fuzzy topological spaces 𝜇 𝑋 , 𝜇 𝑌 and a fuzzy continuous surjection 𝜋: 𝜇 𝑋 → 𝜇 𝑌 . The space 𝜇 𝑋 is called fuzzy total [𝑜𝑟 𝑓𝑢𝑧𝑧𝑦 𝑓𝑖𝑏𝑒𝑟𝑒𝑑] space, 𝜋 is termed the projection, and 𝜇 𝑌 is a fuzzy base space. Then 𝐹 = 𝜋 -1 (𝑦) and F is called fuzzy fiber over 𝑦. We refer to (𝜇 𝑋 , 𝜋, 𝜇 𝑌 ) as a fuzzy fiber structure over 𝜋. fibration if it has fuzzy lifting homotopy property with respect to all (CW-complex 𝜇 𝑌 ) spaces. b) The fuzzy fiber space has fuzzy lifting homotopy property with respect to classes of space which called fuzzy fibration.

Non-trivial of fuzzy principal bundle

In this section, we give examples for the fuzzy projection map which is fuzzy fibration but not a fuzzy Hurewics fibration. fuzzy projection 𝜋: 𝜇 𝑋 → 𝜇 𝑌 , where gives a fuzzy trivial Gbundle over 𝜇 𝑌 since 𝜇 𝑋 | 𝑈 ≃ 𝜇 𝑋 𝑈 , and 𝜇 𝑋 | 𝑉 ≃ 𝜇 𝑋 𝑉 are fuzzy trivial G-bundles. To prove that 𝜋 is not a fuzzy Hurewics fibration. suppose that 𝜇 𝑌 is fuzzy contractible. Also, if 𝜋: 𝜇 𝑋 → 𝜇 𝑌 is fuzzy fibration, the admit a fuzzy section it would be necessary. In particular, let 𝐻 𝑡: 𝜇 𝑌 → 𝜇 𝑌 be a fuzzy null-homotopy of 𝑖𝑑 𝜇 𝑌 . Obviously, we can lift the constant map 𝐻 1 to 𝜇 𝑋 . Suppose that 𝜋 is fuzzy Hurewics fibration, the fuzzy lifting homotopy property then we get a lift 𝐻 *: 𝜇 𝑌 → 𝜇 𝑋 and as a result a section 𝐻 𝑜 * of 𝜋. But we showed previously that 𝜋 admits no fuzzy section. Therefore 𝜋 is not a fuzzy Hurewics fibration. Assume 𝜇 𝐺 = 𝜇 𝐺𝐿 +1 (ℝ) = ( 𝕝 + , +, . ) be the fuzzy topological abelian group which given by the positive reals with multiplication. Let the fuzzy trivial G-bundles over 𝑈 and 𝑉, given by 𝜇 𝑋 𝑈 = 𝑈 × 𝜇 𝐺 and 𝜇 𝑋 𝑉 = 𝑉 × 𝜇 𝐺 respectively. The Fuzzy principal 𝜇 𝐺 -bundle 𝜇 𝑋 over 𝜇 𝑌 whose construct by gluing 𝜇 𝑋 𝑈 and 𝜇 𝑋 𝑉 along 𝑈 ∩ 𝑉 by the 𝜇 𝐺 -isomorphism. ⁄ ∈ 𝜔} indexed by the slicing neighborhoods, called the slicing functions, where each 𝜙 𝑈 , is defined on the subspace 𝑈 × 𝑝 -1 (𝑈) \ $, of the product space 𝑋 × 𝐸, with images in 𝐸 in such a way that the following two conditions are satisfied

𝜑𝑓 𝜇 𝑋 𝑈 |𝑈 ∩ 𝑉

If a slicing structure 𝑆 = {𝜔, 𝜙 𝑈 } for 𝑝 exists, we say that 𝑝: 𝐸 → 𝑋 has the slicing structure property (SSP).

𝐵𝑃 ⇒ 𝑆𝑆𝑃 ⟹ 𝑃𝑎𝑟𝑎𝐶𝐻𝑃

If that 𝑝: 𝐸 → 𝑋 has the SSP then 𝐸 is called sliced fiber space over 𝑋. In addition the triple (𝐸, 𝑝, 𝑋is called a 𝐻𝑢 fibre space (for more details see) [8] . Where {𝑈 𝑉 } cover for 𝜇 𝑌 and {𝜎 𝑉 } is connecting map. 𝑄 𝑉 a fuzzy slicing function so (𝜇 𝑋 , 𝜋, 𝜇 𝑌 ) is a 𝐻𝑢 fuzzy fibre space.

Conclusion

This study successfully introduced and explored the concept of Fuzzy Slicing Structure Property (FuSSP) within the context of fuzzy fiber spaces. By analyzing and proving that fuzzy fibrations do not necessarily qualify as fuzzy Hurewics fibrations, we highlighted a significant distinction in fuzzy topological structures. The research provided key insights into the behaviors of fuzzy fiber bundles, paving the way for future studies in this area. The findings contribute to the broader understanding of fuzzy topology, offering a foundation for potential applications and further theoretical developments in this field.

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