A new kind of pseudo topological (bc) -manifold

Introduction

Topological concepts, such as the concept of the filter, as it is used to describe pseudo topology and its properties, just as series are used to describe topology and its properties was first discussed by Henri Cartan in 1937 [1] . After that it was proposed by Bourbaki, N. in 1940 [2] and then by Choquet in [3] who used it to define convergence space and convergence structures, then after him Gähler in 1976 [4] . Pseudo topological spaces are one of the basic structures in mathematical analysis. The researcher in [5] defined the convergence structures and several concepts related to this space. After that, in [6,7,8,9] , the two researchers studied new properties of this space. In our research, we gave a new definition of the 𝑏𝑐 ̃-open set and relied on it as a basic 𝑏𝑐 ̃-structure to define the 𝑏𝑐 ̃-chart, which in turn is considered the basic building block for building the pseudo 𝑏𝑐 ̃-manifold. The concept introduced in our study can be effectively applied to a specific class of spaces, particularly fuzzy Fréchet spaces; see [10 11, 12, 13] . These works have extensively examined the structural and functional properties of such spaces, providing a solid foundation for further theoretical developments. Building upon these insights, our study offers a new perspective that aligns with and extends the existing results in this area.

Preliminaries

Definition 2.1 [5] A filter in a real vector space 𝑀 is a non-empty set ℋof subsets of 𝑀, such that:

Definition 2.2 [5]: A limit structure (pseudo topology or p. t. ) 𝔍 on 𝑀 is map: (𝔍: 𝑀 → 2 Ω(𝑀) ), 𝑥 ↦ 𝔍(𝑥). If ℋ ∈ 𝔍(𝑥), ℋ converges to 𝑥 in 𝔍 if every set 𝐹 ∈ ℋ contain 𝑥 and written ℋ ↓ 𝑥 (Or ℋ ∈ 𝔍(𝑥))and for all filters ℋ and 𝕡:

Definition 2.3 [5]: A 𝑇 𝑃𝑉𝑆 is both a vector space (v.s.) and p.t. on it (The continuity of addition and multiplication maps). Definition 2.4 [5] Let (ℳ, 𝔍) be a 𝑇 𝑃𝑉𝑆 . The interior 𝒜 of a set 𝒜 ⊆ ℳ is explained by: 𝑖𝑛(𝒜) ={𝑥 ∈ 𝒜: ℋ ∈ 𝔍(𝑥) implies 𝒜 ∈ ℋ}.We call it (open set) if 𝒜 = 𝑖𝑛(𝒜). Definition 2.5 [5]: Let (ℳ , 𝔍) be a 𝑇 𝑃𝑉𝑆 and 𝒜 ⊆ ℳ , then the closure of 𝒜 is 𝒜 ̅ = (CL 𝒜) = {𝑥 ∈ ℳ: ∃ 𝕡 ∈ 𝔍(𝑥), 𝒜 ∈ 𝕡}. We can be called 𝔍-closed set (for easily closed set) if 𝒜 = CL 𝒜.

Main Results

) is composite maps of class 𝐶 ̃∞then its map of class𝐶 ̃∞ on 𝛼 1 (𝑢 1 ∩ 𝑢 2 ∩ 𝑢 3 ), hence (𝑢 1 , 𝛼 1 ) be compatible of class𝐶 ̃∞ with (𝑢 3 , 𝛼 3 ) on 𝛼 1 (𝑢 1 ∩ 𝑢 2 ∩ 𝑢 3 ). Lemma 3.19: Let 𝒜 = {(𝑧 𝑗 , 𝒻 𝑗 )} 𝑗∈𝐽 be 𝐶 ̃∞-𝑏𝑐 ̃-atlas on a 𝒫. 𝑏𝑐 ̃-m. ℳ. If two 𝐶 ̃∞-𝑏𝑐 ̃-charts (𝑢 , 𝜗) , (𝜔, 𝛾) are compatible of 𝐶 ̃∞ with the 𝐶 ̃∞-𝑏𝑐 ̃-atlas, 𝒜 = {(𝑧 𝑗 , 𝒻 𝑗 )} 𝑗∈𝐽 , then they are compatible of 𝐶 ̃∞ with each other.

Proof: Let 𝑧 ∈ 𝑢 ∩ 𝜔. We will prove that 𝛾 ∘ 𝜗 -1 is 𝐶 ̃∞ at 𝜗(𝑧). Since 𝒜 = {(𝑢 𝛿 , 𝜉 𝛿 )} is an 𝐶 ̃∞-𝑏𝑐 ̃-atlas in ℳ, 𝑧 ∈ 𝑢 𝛿 for some 𝛿. Then, 𝑧 ∈ 𝑢 ∩ 𝜔 ∩ 𝑢 𝛿 . From above the proposition, 𝛾 ∘ 𝜗 -1 = (𝛾 ∘ 𝒻 𝛿 -1 ) ∘ (𝒻 𝛿 ∘ 𝜗 -1 ) is smooth on 𝜗(𝑢 ∩ 𝜔 ∩ 𝑢 𝛿 ), hence at 𝜗(𝑧). Since 𝑧 is an arbitrary point of 𝑢 ∩ 𝜔, then 𝛾 ∘ 𝜗 -1 is smooth on 𝜗(𝑢 ∩ 𝜔). Similarly, 𝜗 ∘ 𝛾 -1 is smooth on 𝛾(𝑢 ∩ 𝜔). Therefore, (𝑢 , 𝜗) and (𝜔, 𝛾) are compatible of 𝐶 ̃∞. Proof: 𝑓 ∘ 𝜃 𝑛 -1 is to be proved as smooth for every 𝐶 ̃∞-𝑏𝑐 ̃chart (𝑢, 𝜃) on ℳ. It is enough for us to prove that it is smooth in a neighborhood of any point 𝑦 = 𝜃(𝑥 0 ) ∈ 𝜃(𝑢). For any 𝑦 0 ∈ 𝑢,∃ 𝑏𝑐 ̃-chart (𝑢 𝑛 , 𝜃 𝑛 ) ∈ 𝐶 ̃∞-𝑏𝑐 ̃-atlas

Proposition 3.20:

) is 𝐶 ̃∞in a neighborhood of 𝑦. Proof: Let 𝑠, 𝑟 be a two deferent points in ℳ 1 . Suppose 𝔍 (𝑠) ∩ 𝔍(𝑟) ≠ ∅ then ∃ a filter 𝔉 in 𝔍 (𝑠) ∩ 𝔍( 𝑟) , hence 𝔉 ∈ 𝔍 (𝑠) and 𝔉 ∈ 𝔍 (𝑟) on ℳ 1 . Then, 𝜎(𝔉) ∈ 𝔍 (𝜎(𝑠)) and 𝜎(𝔉) ∈ 𝔍 (𝜎(𝑟)), since ℳ 2 is separated then, 𝜎(𝑠) = 𝜎(𝑟). From infectivity of 𝜎 we get the contradiction. So, 𝔍 (𝑠) ∩ 𝔍( 𝑟) = ∅. So ℳ 1 is separated. Let 𝑠 ∈ ℳ 2 be evenly 𝑏𝑐 ̃-covered neighbourhood. We shrink 𝔲 (if necessary), and we impose it is the domain of a 𝑏𝑐 ̃-coordinate map 𝔔: 𝔲 → 𝑅. If 𝓋 is a component of 𝜎 -1 (𝔲) and 𝔔 ̃= 𝔔 ∘ 𝜎: 𝔲 ̃→ 𝑅, then (𝔲 ̃, 𝔔 ̃) is a 𝑏𝑐 ̃-chart on ℳ 2 . If the two 𝑏𝑐 ̃charts (𝔲 ̃ , 𝔔 ̃) and (𝓋 ̃, 𝜑 ̃) overlap, the transition map it written as 𝜑 ̃∘ 𝔔 ̃-1 = ( 𝜑 ∘ 𝜎/ 𝔲 ̃∩𝓋 ̃) ∘ ( 𝔔 ∘ 𝜎 ∕ 𝔲 ̃∩𝓋 ̃)-1 =𝜑 ∘ 𝜎 ∕ 𝔲 ̃∩𝓋 ̃∘ ( 𝜎 ∕ 𝔲 ̃∩𝓋 ̃)-1 ∘ 𝔔 -1 =𝜑 ∘ 𝔔 -1 which is 𝐶 ̃∞.

Conclusion

The research presented a new type of pseudo-𝑏𝑐 ̃-manifolds after defining the 𝑏𝑐 ̃-open set in the pseudo topological space in a new way. We also dealt with the study of functions defined on this pseudo-manifold and the Euclidean space, homomorphism functions, and then we expanded it to be diffeomorphism functions. We also studied the functions between pseudo-𝑏𝑐 ̃-manifolds, and finally we studied the concepts of 𝑏𝑐 ̃-coverage space and 𝑏𝑐 ̃-coverage functions.

References

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