Theorem of Van Kampen, covering spaces, examples and
... − π1 (Uµ ), which send the class of a loop in the small open subset Uλ to the class of same loop viewed in the larger open subset Uµ : by Proposition 1.5.1 and the subsequent discussion it is then enough to consider the free product of the groups π1 (Uλ )’s of a selected family of open subsets Uλ wi ...
... − π1 (Uµ ), which send the class of a loop in the small open subset Uλ to the class of same loop viewed in the larger open subset Uµ : by Proposition 1.5.1 and the subsequent discussion it is then enough to consider the free product of the groups π1 (Uλ )’s of a selected family of open subsets Uλ wi ...
65, 3 (2013), 419–424 September 2013 TOTALLY BOUNDED ENDOMORPHISMS ON A TOPOLOGICAL RING
... Proof. (i) Suppose two nets (Tα ) and (Sα ) of btb-bounded endomorphisms converge uniformly to zero on bounded sets. Fix a bounded set B ⊆ X. Let W be an arbitrary zero neighborhood in X. Find a zero neighborhood V such that V + V ⊆ W . There are some α0 and α1 with Tα (B) ⊆ V for each α ≥ α0 and Sα ...
... Proof. (i) Suppose two nets (Tα ) and (Sα ) of btb-bounded endomorphisms converge uniformly to zero on bounded sets. Fix a bounded set B ⊆ X. Let W be an arbitrary zero neighborhood in X. Find a zero neighborhood V such that V + V ⊆ W . There are some α0 and α1 with Tα (B) ⊆ V for each α ≥ α0 and Sα ...
Introduction to Quad topological spaces(4-tuple topology)
... Recently the topological structures had a lot of applications in many real life situations. Starting from single topology it extended to bitopology and tritopology with usual definitions. The concept of a bitopological space was first introduced by Kelly [1] and extention to tri-topological spaces w ...
... Recently the topological structures had a lot of applications in many real life situations. Starting from single topology it extended to bitopology and tritopology with usual definitions. The concept of a bitopological space was first introduced by Kelly [1] and extention to tri-topological spaces w ...
paracompactness with respect to anideal
... The following obvious result is stated for the sake of completeness THEOREM IL2. If (X,-,T) is ’-paracompact and 7 is an ideal on X such that Z C_ ,7, then (X,7.,) is J-paracompact. Given a space (X,-,7), the collection/(2-,7.) {U- I:U E 7.,I E 2-} is a basis for a topology -’(Z) v-*, then finer tha ...
... The following obvious result is stated for the sake of completeness THEOREM IL2. If (X,-,T) is ’-paracompact and 7 is an ideal on X such that Z C_ ,7, then (X,7.,) is J-paracompact. Given a space (X,-,7), the collection/(2-,7.) {U- I:U E 7.,I E 2-} is a basis for a topology -’(Z) v-*, then finer tha ...
INTRODUCTION TO ALGEBRAIC TOPOLOGY 1.1. Topological
... maps. Explicitly HomTop (X, Y ) is the set of continuous maps from X to Y and composition is a set map ◦ : HomTop (Y, Z) × HomTop (X, Y ) → HomTop (X, Z). These satisfy the Axioms of a category: the existence and properties of identity morphisms IdX ∈ HomTop (X, X) and associativity of composition o ...
... maps. Explicitly HomTop (X, Y ) is the set of continuous maps from X to Y and composition is a set map ◦ : HomTop (Y, Z) × HomTop (X, Y ) → HomTop (X, Z). These satisfy the Axioms of a category: the existence and properties of identity morphisms IdX ∈ HomTop (X, X) and associativity of composition o ...
