On z-θ-Open Sets and Strongly θ-z
... (a) f is strongly θ-z-continuous, (b) For each x ∈ X and each open set V of Y containing f (x), there exists U ∈ zR(X, x) such that f [U] ⊂ V , (c) f −1 [V ] is z-θ-open in X for each open set V of Y , (d) f −1 [F ] is z-θ-closed in X for each closed set F of Y , (e) f [z-clθ (A)] ⊂ cl(f [A]) for ea ...
... (a) f is strongly θ-z-continuous, (b) For each x ∈ X and each open set V of Y containing f (x), there exists U ∈ zR(X, x) such that f [U] ⊂ V , (c) f −1 [V ] is z-θ-open in X for each open set V of Y , (d) f −1 [F ] is z-θ-closed in X for each closed set F of Y , (e) f [z-clθ (A)] ⊂ cl(f [A]) for ea ...
AROUND EFFROS` THEOREM 1. Introduction. In 1965 when Effros
... metric the authors did not expect any essentially new results: they consider this concept only as a comfortable language to describe some properties connected with the homogeneity of spaces. The paper consists of six sections. §2, which plays an auxiliary role, contains an evaluation of the Borel cl ...
... metric the authors did not expect any essentially new results: they consider this concept only as a comfortable language to describe some properties connected with the homogeneity of spaces. The paper consists of six sections. §2, which plays an auxiliary role, contains an evaluation of the Borel cl ...
Vasile Alecsandri” University of Bac˘au Faculty of Sciences Scientific
... Corollary 4.2. For a function f : (X, τ ) → (Y, σ), the following properties are equivalent: (1) f is closed; (2) f is g-closed and LC-closed; (3) f is sg ∗ -closed and slc-closed; (4) f is pg ∗ -closed and plc-closed; (5) f is αg ∗ -closed and αlc-closed; (6) f is βg ∗ -closed and a βlc-set. Proof. ...
... Corollary 4.2. For a function f : (X, τ ) → (Y, σ), the following properties are equivalent: (1) f is closed; (2) f is g-closed and LC-closed; (3) f is sg ∗ -closed and slc-closed; (4) f is pg ∗ -closed and plc-closed; (5) f is αg ∗ -closed and αlc-closed; (6) f is βg ∗ -closed and a βlc-set. Proof. ...
On totally − Continuous functions in supra topological spaces
... –continuous if it has the above property at each point of X. Theorem: 2.2 The following statements are equivalent for a function ...
... –continuous if it has the above property at each point of X. Theorem: 2.2 The following statements are equivalent for a function ...
2.2 The abstract Toeplitz algebra
... Proof. Let us show that the canonical map πq : Tq Ñ C pSq q sending sq to Sq is a -isomorphism. The theorem will then follow by Proposition 2.2.12. By Coburn's theorem, we may suppose that q ¡ 0. By the non-commutative Gelfand-Neumark theorem, we may also assume that Tq B pHq for some Hilbert ...
... Proof. Let us show that the canonical map πq : Tq Ñ C pSq q sending sq to Sq is a -isomorphism. The theorem will then follow by Proposition 2.2.12. By Coburn's theorem, we may suppose that q ¡ 0. By the non-commutative Gelfand-Neumark theorem, we may also assume that Tq B pHq for some Hilbert ...
![arXiv:0903.2024v3 [math.AG] 9 Jul 2009](http://s1.studyres.com/store/data/016334314_1-5f7029642e57766ba62e6f9f2c64e435-300x300.png)
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.