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... closed sets has been studied extensively in recent years by many topologist because generalized closed sets are the only natural generalization of closed sets. More importantly, they also suggest several new properties of topological spaces. Maki [14] introduced the notion of Λ-sets in topological s ...
... closed sets has been studied extensively in recent years by many topologist because generalized closed sets are the only natural generalization of closed sets. More importantly, they also suggest several new properties of topological spaces. Maki [14] introduced the notion of Λ-sets in topological s ...
Somewhat continuous functions
... LEVINE defines in [3] the notion of a D-space. A topological space (Z, 5") is said to be a D-space provided every nonempty open subset of X is dense in X. Theorem 19. / / / : {X, ^) -^ (F, ^) is a somewhat continuous function from X onto Yand X is a D-space, then Y is a D-space. Theorem 20. Suppose ...
... LEVINE defines in [3] the notion of a D-space. A topological space (Z, 5") is said to be a D-space provided every nonempty open subset of X is dense in X. Theorem 19. / / / : {X, ^) -^ (F, ^) is a somewhat continuous function from X onto Yand X is a D-space, then Y is a D-space. Theorem 20. Suppose ...
Far East Journal of Mathematical Sciences (FJMS)
... denoted by E ( F ) is the set of points ( x, y ) with x, y ∈ F satisfying (8) together with a point o, called the point at infinity. Equation (8) is called the Weierstrass equation for E, and Δ( E ( F )) = − 16(4 g 23 + 27 g 32 ) is known as the discriminant of E ( F ). ...
... denoted by E ( F ) is the set of points ( x, y ) with x, y ∈ F satisfying (8) together with a point o, called the point at infinity. Equation (8) is called the Weierstrass equation for E, and Δ( E ( F )) = − 16(4 g 23 + 27 g 32 ) is known as the discriminant of E ( F ). ...
SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A — Part 2 II
... SOLUTION. In order to define a topological space it is enough to define the family F of closed subsets that satisfies the standard properties: It contains the empty set and Y , it is closed under taking arbitrary intersections, and it is closed under taking the unions of two subsets. If we are given ...
... SOLUTION. In order to define a topological space it is enough to define the family F of closed subsets that satisfies the standard properties: It contains the empty set and Y , it is closed under taking arbitrary intersections, and it is closed under taking the unions of two subsets. If we are given ...
3-manifold
![](https://commons.wikimedia.org/wiki/Special:FilePath/3-Manifold_3-Torus.png?width=300)
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.