Introduction to Topology
... – Let’s just check for two subsets U1 , U2 first. For each x ∈ U1 ∩ U2 , there are B1 , B2 ∈ B such that x ∈ B1 ⊂ U1 and x ∈ B2 ⊂ U2 . This is because U1 , U2 ∈ TB and x ∈ U1 , x ∈ U2 . By (B2), there is B3 ∈ B such that x ∈ B3 ⊂ B1 ∩ B2 . Now we found B3 ∈ B such that x ∈ B3 ⊂ U. – We can generali ...
... – Let’s just check for two subsets U1 , U2 first. For each x ∈ U1 ∩ U2 , there are B1 , B2 ∈ B such that x ∈ B1 ⊂ U1 and x ∈ B2 ⊂ U2 . This is because U1 , U2 ∈ TB and x ∈ U1 , x ∈ U2 . By (B2), there is B3 ∈ B such that x ∈ B3 ⊂ B1 ∩ B2 . Now we found B3 ∈ B such that x ∈ B3 ⊂ U. – We can generali ...
CATEGORICAL PROPERTY OF INTUITIONISTIC TOPOLOGICAL
... After Atanassov [1, 2, 4] introduced the concept of “intuitionistic fuzzy sets” as a generalization of fuzzy sets, it becomes a popular topic of investigation in the fuzzy set community. Many mathematical advantages of intuitionistic fuzzy sets are discussed in [3]. Çoker [7] generalized topologica ...
... After Atanassov [1, 2, 4] introduced the concept of “intuitionistic fuzzy sets” as a generalization of fuzzy sets, it becomes a popular topic of investigation in the fuzzy set community. Many mathematical advantages of intuitionistic fuzzy sets are discussed in [3]. Çoker [7] generalized topologica ...
important result of the fuzzy tychonoff theorem and
... Tychonoff theorems and initial and final fuzzy topology.fuzzy Tychonoff theorems is describes as – “ Arbitrary products of compact spaces are compact”. We show that with our notion of fuzzy compactness the Tychonoff product theorem is safeguarded. ...
... Tychonoff theorems and initial and final fuzzy topology.fuzzy Tychonoff theorems is describes as – “ Arbitrary products of compact spaces are compact”. We show that with our notion of fuzzy compactness the Tychonoff product theorem is safeguarded. ...
On theta-precontinuous functions
... if the preimage f −1 (V ) of each open set V of Y is preopen in X. Precontinuity was called near continuity by Pták [26] and also called almost continuity by Frolík [9] and Husain [10]. In 1985, Janković [12] introduced almost weak continuity as a weak form of precontinuity. Popa and Noiri [23] int ...
... if the preimage f −1 (V ) of each open set V of Y is preopen in X. Precontinuity was called near continuity by Pták [26] and also called almost continuity by Frolík [9] and Husain [10]. In 1985, Janković [12] introduced almost weak continuity as a weak form of precontinuity. Popa and Noiri [23] int ...
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.