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Redalyc.On a class of ay-open sets in a topological space
... ∈ αγCl({x}) and z ∈ A ⊆ U. It follows from Theorem 2.23, that U ∩ {x} ≠ φ, hence x ∈ U, this implies αγCl(A) ⊆ U. Therefore A is αγ-g.closed. Conversely, suppose that x ∈ αγCl(A) such that αγCl({x}) ∩ A = φ. Since, αγCl({x}) is αγ-closed, therefore X\αγCl({x}) is an αγ-open set in X. Since A ⊆ X\(αγ ...
... ∈ αγCl({x}) and z ∈ A ⊆ U. It follows from Theorem 2.23, that U ∩ {x} ≠ φ, hence x ∈ U, this implies αγCl(A) ⊆ U. Therefore A is αγ-g.closed. Conversely, suppose that x ∈ αγCl(A) such that αγCl({x}) ∩ A = φ. Since, αγCl({x}) is αγ-closed, therefore X\αγCl({x}) is an αγ-open set in X. Since A ⊆ X\(αγ ...
Geometry Proofs
... Definition: Two lines are parallel if and only if they do not intersect. Theorem: If two distinct lines intersect, then they intersect in exactly one point. Postulate (Ruler): There is a one to one correspondence between real numbers and points on a line. Postulate (Protractor): Given line AB and th ...
... Definition: Two lines are parallel if and only if they do not intersect. Theorem: If two distinct lines intersect, then they intersect in exactly one point. Postulate (Ruler): There is a one to one correspondence between real numbers and points on a line. Postulate (Protractor): Given line AB and th ...
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.