on α-i alexandroff spaces
... (1) X is T1 and α-I-Alexandorff. (2) X is discrete. Proof. (1) ⇒ (2) For each x ∈ X and each y = x, there exists an open set Ux containing x such that y ∉ Ux (X is T1 ). Since X is αI-Alexandorff and since {x} = ∩y=x Ux , then {x} is αI-open and hence open, since a singleton is α-I-open if and only ...
... (1) X is T1 and α-I-Alexandorff. (2) X is discrete. Proof. (1) ⇒ (2) For each x ∈ X and each y = x, there exists an open set Ux containing x such that y ∉ Ux (X is T1 ). Since X is αI-Alexandorff and since {x} = ∩y=x Ux , then {x} is αI-open and hence open, since a singleton is α-I-open if and only ...
A Discourse on Analytical Study of Nearly
... Every p-open set is a β open set and every regular- open set is a α-open set, but converse is not true. Every α-open set is p-open as well as s-open. The α-sets with respect to a given topology are exactly those sets which are difference between an open set ...
... Every p-open set is a β open set and every regular- open set is a α-open set, but converse is not true. Every α-open set is p-open as well as s-open. The α-sets with respect to a given topology are exactly those sets which are difference between an open set ...
a first countable space is a topological space in which there exist a
... Definition: a topological space (X,T) is said to be second countable space iff there exist a countable base for T. Example : (R,U) is second countable space. Example : (R,D) is not second countable space. Theorem: every second countable space is first countable space Theorem: the property of being ...
... Definition: a topological space (X,T) is said to be second countable space iff there exist a countable base for T. Example : (R,U) is second countable space. Example : (R,D) is not second countable space. Theorem: every second countable space is first countable space Theorem: the property of being ...
10012555.5 Properties of Parallelograms
... make sure that the lines are perfectly straight. Use the lines on the grid to make sure that the lines are perfectly parallel. Do not draw a rectangle or a square. Label your parallelogram LOVE to indicate how you feel about math. ...
... make sure that the lines are perfectly straight. Use the lines on the grid to make sure that the lines are perfectly parallel. Do not draw a rectangle or a square. Label your parallelogram LOVE to indicate how you feel about math. ...
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.