g-COMPACTNESS LIKE PROPERTIES IN GENERALIZED
... Theorem 5.13. g-pre compact subset of a generalized topological space is g-β-compact if either one of the following holds (i) g-interior of each g-closed set is g-closed (ii) g-closure of each g-open set is g-open. Proof. Let A be a g-pre compact subset of a generalized topological space X and let V ...
... Theorem 5.13. g-pre compact subset of a generalized topological space is g-β-compact if either one of the following holds (i) g-interior of each g-closed set is g-closed (ii) g-closure of each g-open set is g-open. Proof. Let A be a g-pre compact subset of a generalized topological space X and let V ...
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Applications of some strong set-theoretic axioms to locally compact
... this, we recall a closely related concept which was introduced in [Ny1]: 1.8. Definition. A space X is a Type I space if it is the union of an ω1 -sequence hXα : α < ω1 i of open subspaces such that X α ⊂ Xβ whenever α < β and such that X α is Lindelöf for all α. SuchSan ω1 -sequence will be called ...
... this, we recall a closely related concept which was introduced in [Ny1]: 1.8. Definition. A space X is a Type I space if it is the union of an ω1 -sequence hXα : α < ω1 i of open subspaces such that X α ⊂ Xβ whenever α < β and such that X α is Lindelöf for all α. SuchSan ω1 -sequence will be called ...
DISCONTINUOUS GROUPS AND CLIFFORD
... Definition 1.3.1. The action of Γ on X is said to be: i) properly discontinuous if ΓS is a finite subset for any compact subset S of X, ii) free if Γ{p} is trivial for any p ∈ X. Then we have the following standard fact: Lemma 1.3.2. Suppose that a discrete group Γ acts on a [C ∞ , Riemannian, compl ...
... Definition 1.3.1. The action of Γ on X is said to be: i) properly discontinuous if ΓS is a finite subset for any compact subset S of X, ii) free if Γ{p} is trivial for any p ∈ X. Then we have the following standard fact: Lemma 1.3.2. Suppose that a discrete group Γ acts on a [C ∞ , Riemannian, compl ...
Lecture Notes (unique pdf file)
... ⇒ Let (X, τ ) be a topological space, x ∈ X and F(x) the filter of neighbourhoods of x. Then (N1) trivially holds by definition of neighbourhood of x. To show (N2), let us take A ∈ F(x). Since A is a neighbourhood of x, there exists B ∈ τ s.t. x ∈ B ⊆ A. Then clearly B ∈ F(x). Moreover, since for an ...
... ⇒ Let (X, τ ) be a topological space, x ∈ X and F(x) the filter of neighbourhoods of x. Then (N1) trivially holds by definition of neighbourhood of x. To show (N2), let us take A ∈ F(x). Since A is a neighbourhood of x, there exists B ∈ τ s.t. x ∈ B ⊆ A. Then clearly B ∈ F(x). Moreover, since for an ...
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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.