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... exists a preopen set ρ in X containing xε such that f (ρ) ≤ β. Therefore, we obtain yν ∈ µ and f (ρ) ∧ µ = ∅. This shows that G(f ) is fuzzy pre-co-closed. Theorem 28. If f : X → Y is fuzzy precontinuous and Y is fuzzy co-T1 , then G(f ) is fuzzy pre-co-closed in X × Y . Proof. Let (xε , yν ) ∈ (X × ...

... exists a preopen set ρ in X containing xε such that f (ρ) ≤ β. Therefore, we obtain yν ∈ µ and f (ρ) ∧ µ = ∅. This shows that G(f ) is fuzzy pre-co-closed. Theorem 28. If f : X → Y is fuzzy precontinuous and Y is fuzzy co-T1 , then G(f ) is fuzzy pre-co-closed in X × Y . Proof. Let (xε , yν ) ∈ (X × ...

FUZZY SEMI α-IRRESOLUTE FUNCTIONS 1. Introduction The fuzzy

... introduced and developed by Chang [4] and since then various notions in classical topology have been extended to fuzzy topological spaces. The concept of semi αirresolute functions was introduced in [8]. In this paper we introduce and study several interesting properties of the fuzzy version of semi ...

... introduced and developed by Chang [4] and since then various notions in classical topology have been extended to fuzzy topological spaces. The concept of semi αirresolute functions was introduced in [8]. In this paper we introduce and study several interesting properties of the fuzzy version of semi ...

Gδ–SEPARATION AXIOMS IN ORDERED FUZZY TOPOLOGICAL

... Gδ -neighbourhood of (x, y) ∈ X × X. Hence we can find a fuzzy Gδ -set λ × µ such that λ × µ < (1 − χG ) and λ is fuzzy Gδ -set such that λ(x) > 0 and µ is a fuzzy Gδ -set such that µ(y) > 0. We now claim that Iσ0 (λ) ∧ Dσ0 (µ) = 0. For if z ∈ X is such that (Iσ0 (λ) ∧ > 0, then Iσ0 (λ)(z) ∧ Dσ0 (µ) ...

... Gδ -neighbourhood of (x, y) ∈ X × X. Hence we can find a fuzzy Gδ -set λ × µ such that λ × µ < (1 − χG ) and λ is fuzzy Gδ -set such that λ(x) > 0 and µ is a fuzzy Gδ -set such that µ(y) > 0. We now claim that Iσ0 (λ) ∧ Dσ0 (µ) = 0. For if z ∈ X is such that (Iσ0 (λ) ∧ > 0, then Iσ0 (λ)(z) ∧ Dσ0 (µ) ...

# 3-manifold

In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. Intuitively, a 3-manifold can be thought of as a possible shape of the universe. Just like a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.