FULL TEXT - RS Publication
... Remark.1.2 [2]:1)and X are g-open in any topological space (X,).2)Every closed sets is gclosed but the converse is not true 3)Union of two g-closed sets is g-closed.4) Arbitrary union of g-closed sets need not be g-closed4) Intersection of g-closed sets need not be g-closed. GO(X) is not a topolo ...
... Remark.1.2 [2]:1)and X are g-open in any topological space (X,).2)Every closed sets is gclosed but the converse is not true 3)Union of two g-closed sets is g-closed.4) Arbitrary union of g-closed sets need not be g-closed4) Intersection of g-closed sets need not be g-closed. GO(X) is not a topolo ...
“Scattered spaces”
... that X α = X α+1 . This ordinal α, denoted by ht(X), is called the CantorBendixson height, or the height of X. Clearly the subspace X (α) does not have any isolated points, it is dense-in-itself. The derived sets are all closed, so X (α) is perfect. Moreover, Y = X \ X (α) is scattered and so it has ...
... that X α = X α+1 . This ordinal α, denoted by ht(X), is called the CantorBendixson height, or the height of X. Clearly the subspace X (α) does not have any isolated points, it is dense-in-itself. The derived sets are all closed, so X (α) is perfect. Moreover, Y = X \ X (α) is scattered and so it has ...
Splitting of short exact sequences for modules
... In Section 2 we will give two ways to characterize when a short exact sequence of Rmodules splits. Section 3 will discuss a few consequences. Before doing that, we want to stress that being split is not just saying there is an isomorphism M → N ⊕ P of Rmodules, but how the isomorphism works with the ...
... In Section 2 we will give two ways to characterize when a short exact sequence of Rmodules splits. Section 3 will discuss a few consequences. Before doing that, we want to stress that being split is not just saying there is an isomorphism M → N ⊕ P of Rmodules, but how the isomorphism works with the ...
On topological groups via a-local functions - RiuNet
... On topological groups via a-local functions Wadei Al-Omeri a , Mohd. Salmi Md. Noorani a and Ahmad. ...
... On topological groups via a-local functions Wadei Al-Omeri a , Mohd. Salmi Md. Noorani a and Ahmad. ...
Covering space
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below. In this case, C is called a covering space and X the base space of the covering projection. The definition implies that every covering map is a local homeomorphism.Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group. An important application comes from the result that, if X is a ""sufficiently good"" topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of X and the conjugacy classes of subgroups of the fundamental group of X.