Universitat Jaume I Departament de Matem` atiques BOUNDED SETS IN TOPOLOGICAL
... A boundedness structure (bornology) on a topological space is an ideal of subsets (that is, closed under taking subsets and unions of finitely many elements) containing all singletons. In this paper, we introduce certain functions (boundedness maps) as a tool in order to deal with the global propert ...
... A boundedness structure (bornology) on a topological space is an ideal of subsets (that is, closed under taking subsets and unions of finitely many elements) containing all singletons. In this paper, we introduce certain functions (boundedness maps) as a tool in order to deal with the global propert ...
Lecture notes of Dr. Hicham Gebran
... Examples. 1) [a, b] is closed because its complement ]∞, a[∪]b, ∞[ is open. 2) [a, ∞[ and ] − ∞, a] are closed. 3) In a metric space singletons are closed. Indeed, let {a} be a singleton (one point set) and let x ∈ X − {a}. Then x 6= a and so d(x, a) > 0. We claim that B(x, d(x, a)) ⊂ X − {a}. Indee ...
... Examples. 1) [a, b] is closed because its complement ]∞, a[∪]b, ∞[ is open. 2) [a, ∞[ and ] − ∞, a] are closed. 3) In a metric space singletons are closed. Indeed, let {a} be a singleton (one point set) and let x ∈ X − {a}. Then x 6= a and so d(x, a) > 0. We claim that B(x, d(x, a)) ⊂ X − {a}. Indee ...
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... begin with two disjoint subsets C; D R and for each x 2 D a sequence hxn i in C converging to x. They let X(C; D) be the union C [ D but with points of C isolated and neighbourhoods of points of D containing tails of the corresponding sequences. The essential features of X(C; D) are then preserved ...
... begin with two disjoint subsets C; D R and for each x 2 D a sequence hxn i in C converging to x. They let X(C; D) be the union C [ D but with points of C isolated and neighbourhoods of points of D containing tails of the corresponding sequences. The essential features of X(C; D) are then preserved ...
![arXiv:math/0607274v2 [math.GT] 21 Jun 2007](http://s1.studyres.com/store/data/004624723_1-689e7c37d19114c6d0cb2b7e64bf5915-300x300.png)
LOCALIZATION OF ALGEBRAS OVER COLOURED OPERADS
... A common feature of these examples is that they can be described in terms of algebras over operads or, in some cases, algebras over coloured operads. Coloured operads first appeared in the book of Boardman and Vogt [BV73] on homotopy invariant algebraic structures on topological spaces. They can be ...
... A common feature of these examples is that they can be described in terms of algebras over operads or, in some cases, algebras over coloured operads. Coloured operads first appeared in the book of Boardman and Vogt [BV73] on homotopy invariant algebraic structures on topological spaces. They can be ...
Some Properties of Contra-b-Continuous and Almost Contra
... Proof. Let V be any regular closed set on Y . Then since f is almost contra-bcontinuous and almost continuous, then by Theorem 3.9 f is R-map. Hence f −1 (V ) is regular closed in X . Let {Vα : α ∈ I} be any regular closed cover of Y . Then { f −1 (Vα ) : α ∈ I} is a regular closed cover of X and si ...
... Proof. Let V be any regular closed set on Y . Then since f is almost contra-bcontinuous and almost continuous, then by Theorem 3.9 f is R-map. Hence f −1 (V ) is regular closed in X . Let {Vα : α ∈ I} be any regular closed cover of Y . Then { f −1 (Vα ) : α ∈ I} is a regular closed cover of X and si ...