A unified theory of weakly contra-(µ, λ)
... 4. Weak contra-(µ, λ)-continuity and (gµ, λ)-continuity Definition 4.1. Let (X, τ ) be a topological space. A subset A of X is said to be 1. g-closed [41] if Cl(A) ⊂ U whenever A ⊂ U and U ∈ τ , 2. αg-closed [27] if αCl(A) ⊂ U whenever A ⊂ U and U ∈ τ , 3. gs-closed [26] if sCl(A) ⊂ U whenever A ⊂ U ...
... 4. Weak contra-(µ, λ)-continuity and (gµ, λ)-continuity Definition 4.1. Let (X, τ ) be a topological space. A subset A of X is said to be 1. g-closed [41] if Cl(A) ⊂ U whenever A ⊂ U and U ∈ τ , 2. αg-closed [27] if αCl(A) ⊂ U whenever A ⊂ U and U ∈ τ , 3. gs-closed [26] if sCl(A) ⊂ U whenever A ⊂ U ...
Topology Proceedings
... function from X to Y is de Groot if it is pairwise continuous from (X, iX, i~) to (Y, ry, I?). If we denote r V r G by i SG then clearly the de Groot maps are SG-continuous. Two examples of the de Groot dual are particularly impor tant in this paper: If X is finite, then each of its subsets is comp ...
... function from X to Y is de Groot if it is pairwise continuous from (X, iX, i~) to (Y, ry, I?). If we denote r V r G by i SG then clearly the de Groot maps are SG-continuous. Two examples of the de Groot dual are particularly impor tant in this paper: If X is finite, then each of its subsets is comp ...
Fibrewise Compactly
... ogy used in [6], which I hope is largely self-explanatory, will be adopted here, except that it is convenient to follow the usage of Bourbaki and include the fibrewise Hausdorff condition in the definition of the terms fibrewise compact and fibrewise locally compact. §1. The Retraction Functor We wo ...
... ogy used in [6], which I hope is largely self-explanatory, will be adopted here, except that it is convenient to follow the usage of Bourbaki and include the fibrewise Hausdorff condition in the definition of the terms fibrewise compact and fibrewise locally compact. §1. The Retraction Functor We wo ...
Sequences and nets in topology
... X \ A that converges to y ∈ A. Then (xn ) has a subsequence in X \ A that must still converge to y ∈ A, so A is not sequentially open. Proposition 7. The following are equivalent for any topological space X: 1. X is sequential; 2. for any topological space Y and function f : X → Y , f is continuous ...
... X \ A that converges to y ∈ A. Then (xn ) has a subsequence in X \ A that must still converge to y ∈ A, so A is not sequentially open. Proposition 7. The following are equivalent for any topological space X: 1. X is sequential; 2. for any topological space Y and function f : X → Y , f is continuous ...
BOREL SETS, WELL-ORDERINGS OF R AND THE CONTINUUM
... (X, T ), where T is the topology with open basis B(x, r) = {y ∈ X | d(x, y) < r} x ∈ X, r > 0. In this case, we say that the metric d is compatible with the topology T and we also say that the topology T is metrizable. Definition 2.2. A topological space X is said to be Hausdorff iff for all x 6= y ...
... (X, T ), where T is the topology with open basis B(x, r) = {y ∈ X | d(x, y) < r} x ∈ X, r > 0. In this case, we say that the metric d is compatible with the topology T and we also say that the topology T is metrizable. Definition 2.2. A topological space X is said to be Hausdorff iff for all x 6= y ...
topologies on spaces of subsets
... 5.8). Typical among these is that the function a:zA(zA(X)) —»cvf(X), which maps a collection of sets into its union, is continuous (Theorems 5.7.1 and 5.7.2). Next we study the relationships between a function/:X —> Fand the function it induces among the hyperspaces (Theorem 5.10). We conclude this ...
... 5.8). Typical among these is that the function a:zA(zA(X)) —»cvf(X), which maps a collection of sets into its union, is continuous (Theorems 5.7.1 and 5.7.2). Next we study the relationships between a function/:X —> Fand the function it induces among the hyperspaces (Theorem 5.10). We conclude this ...
AN APPLICATION OF MACKEY`S SELECTION LEMMA 1
... Therefore the map (r, d) is open Corollary 1. Let G be a locally compact groupoid having open range map. Let F be a subset of G(0) meeting each orbit exactly once. If the map dF : GF → G(0) , dF (x) = d (x), is open, then the orbit space G(0) /G is proper. Proof. The fact that G(0) /G is a proper sp ...
... Therefore the map (r, d) is open Corollary 1. Let G be a locally compact groupoid having open range map. Let F be a subset of G(0) meeting each orbit exactly once. If the map dF : GF → G(0) , dF (x) = d (x), is open, then the orbit space G(0) /G is proper. Proof. The fact that G(0) /G is a proper sp ...
