Compact Gδ Sets - College of William and Mary Math Department
... In case C is the collection of all closed subsets of X, one obtains the well-known class of semistratifiable spaces introduced by Creede [13]. In case C is the collection of all compact subsets of X, one has the class of all c-semi-stratifiable (CSS) spaces introduced by H. Martin [25]. In Sections ...
... In case C is the collection of all closed subsets of X, one obtains the well-known class of semistratifiable spaces introduced by Creede [13]. In case C is the collection of all compact subsets of X, one has the class of all c-semi-stratifiable (CSS) spaces introduced by H. Martin [25]. In Sections ...
Jan van MILL and Alexander SCHRIJVER Often, an important: class
... orderable spaces are due to De Groot [ 1 l] and De Groot and Schnare [ 14 !. An idea of De Groot was to represent a supercompact space with binary su by the graph with verte:c set 9’ and an edge between So and Si in 9 if and only if SOf-JS1# 0. De Groat [ 121 proved that the space is completely dete ...
... orderable spaces are due to De Groot [ 1 l] and De Groot and Schnare [ 14 !. An idea of De Groot was to represent a supercompact space with binary su by the graph with verte:c set 9’ and an edge between So and Si in 9 if and only if SOf-JS1# 0. De Groat [ 121 proved that the space is completely dete ...
SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A — Part 2 II
... Given an arbitrary set Y and a operation CL assigning to each subset B ⊂ Y another subset CL(B) ⊂ Y such that (C1) – (C4) all hold, prove that there is a unique topology T on Y such that for all B ⊂ Y , the set CL(B) is the closure of B with respect to T. SOLUTION. In order to define a topological ...
... Given an arbitrary set Y and a operation CL assigning to each subset B ⊂ Y another subset CL(B) ⊂ Y such that (C1) – (C4) all hold, prove that there is a unique topology T on Y such that for all B ⊂ Y , the set CL(B) is the closure of B with respect to T. SOLUTION. In order to define a topological ...
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... continuous linear mapping from E into F with dense range. Then F is WCG. There are some comments in order before we proceed to prove Theorems 3.1 and 3.2. Firstly, recall that a compact Hausdorff space Ω is said to be Eberlein compact whenever Ω is homeomorphic to a weakly compact subset of some Ban ...
... continuous linear mapping from E into F with dense range. Then F is WCG. There are some comments in order before we proceed to prove Theorems 3.1 and 3.2. Firstly, recall that a compact Hausdorff space Ω is said to be Eberlein compact whenever Ω is homeomorphic to a weakly compact subset of some Ban ...
Outline - Durham University
... Proposition 2.7. Every line on S 2 intersects every other line in exactly two antipodal points. Definition 2.8. By the angle between two lines we mean an angle between the corresponding planes: if li = αi ∩ S 2 , i = 1, 2 then ∠(l1 , l2 ) := ∠(α1 , α2 ). Equivalently, ∠(l1 , l2 ) is the angle betwee ...
... Proposition 2.7. Every line on S 2 intersects every other line in exactly two antipodal points. Definition 2.8. By the angle between two lines we mean an angle between the corresponding planes: if li = αi ∩ S 2 , i = 1, 2 then ∠(l1 , l2 ) := ∠(α1 , α2 ). Equivalently, ∠(l1 , l2 ) is the angle betwee ...
The weights of closed subgroups of a locally compact group
... able, that is, its p-rank card A is uncountable and agrees with the p-rank of D.p/ (see [9, p. 656, Corollary A1.36 (iii)]). In view of D.p/ Š .Z.p 1 /.card A/ / by [9, p. 659, Theorem A1.42 (iii)], we can find a direct summand D1 of D.p/ of p-rank @, giving us a direct summand of tor D and thus yie ...
... able, that is, its p-rank card A is uncountable and agrees with the p-rank of D.p/ (see [9, p. 656, Corollary A1.36 (iii)]). In view of D.p/ Š .Z.p 1 /.card A/ / by [9, p. 659, Theorem A1.42 (iii)], we can find a direct summand D1 of D.p/ of p-rank @, giving us a direct summand of tor D and thus yie ...
7 Complete metric spaces and function spaces
... We shall use the characterization above to study compact subsets of spaces of continuous funtions. We make use of the following notion, often easier to check than total boundedness: Definition 7.24. Let X be a topologcal space, (Y, d) be a metric space. A subset F ⊂ C(X, Y) is said to be equicontinu ...
... We shall use the characterization above to study compact subsets of spaces of continuous funtions. We make use of the following notion, often easier to check than total boundedness: Definition 7.24. Let X be a topologcal space, (Y, d) be a metric space. A subset F ⊂ C(X, Y) is said to be equicontinu ...
Chapter II. Continuity
... X consisting of solutions of the system of inequalities f1 (x) ≥ 0, . . . , fn (x) ≥ 0 is closed, while the set consisting of solutions of the system of inequalities f1 (x) > 0, . . . , fn (x) > 0 is open. 9.27. Where in 9.O and 9.P a finite system can be replaced by an infinite one? 9.28. Prove tha ...
... X consisting of solutions of the system of inequalities f1 (x) ≥ 0, . . . , fn (x) ≥ 0 is closed, while the set consisting of solutions of the system of inequalities f1 (x) > 0, . . . , fn (x) > 0 is open. 9.27. Where in 9.O and 9.P a finite system can be replaced by an infinite one? 9.28. Prove tha ...
Regular Generalized Star b-Sets in Bigeneralized
... Tibanga, Iligan City, Philippines c 2014 Josephine Josol Baculta and Helen Moso Rara. This is an open Copyright access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is pro ...
... Tibanga, Iligan City, Philippines c 2014 Josephine Josol Baculta and Helen Moso Rara. This is an open Copyright access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is pro ...
On Kolmogorov Topological Spaces 1
... characteristics of such spaces: X is a Kolmogorov space iff for every pair of distinct points x, y ∈ X the closures {x} and {y} are distinct. There is also reviewed analogous facts on Kolmogorov subspaces of topological spaces. In the presented approach T0 -subsets are introduced and some of their p ...
... characteristics of such spaces: X is a Kolmogorov space iff for every pair of distinct points x, y ∈ X the closures {x} and {y} are distinct. There is also reviewed analogous facts on Kolmogorov subspaces of topological spaces. In the presented approach T0 -subsets are introduced and some of their p ...
Free Topological Groups - Universidad Complutense de Madrid
... This is a good point to turn back to the problem of the existence of free topological groups. Let us consider the non-Abelian case first. It follows from Definition 1.1 that the topology of the group F (X ) (when the latter exists) is maximal in some sense. Here is the exact mathematical formulation ...
... This is a good point to turn back to the problem of the existence of free topological groups. Let us consider the non-Abelian case first. It follows from Definition 1.1 that the topology of the group F (X ) (when the latter exists) is maximal in some sense. Here is the exact mathematical formulation ...
