#8Need part i and ii which yieldswith ß = 1/α'.Pageα′||x||2 ≤ ||x||1, for x ¤ E,||x||2 ≤ ß||x||1, for x ¤ E,55of 204 NeighborhoodDefinition 4.1 (Neighborhood). Let X be a topological space. A neighborhood V c X of x € X isany subset of X for which we can find an open subset U with x ¤ U © V.We will denote by V(x) the set of neighborhoods of x.Though the following proposition is easy, it is nonetheless very useful.Proposition 4.2. A subset U of the topological space X is open if and only if it is a neighborhoodof each of its point.Proof. Exercise.The following properties of neighborhoods are easy to check.Proposition 4.3. If x is a point of the topological space X, we haveZOOM + Exercise 8. Prove the following properties of neighborhoods:Proposition. If x is a point of the topological space X, we have(i) if V' Ɔ V, with V = V(x), then V' = V(x);(ii) if V, V' ≤ V(x), then VÑV' = V(x).

Answered: Image /qna-images/answer/ade56d87-e2c8-4d1b-945d-8ea74f9c76d2.jpg

Answered: Exercise 8. Prove the following…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.2: Vector Spaces

Problem 38E: Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1)...

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Exercise 8. Prove the following properties of neighborhoods: Proposition. If x is a point of the topological space X, we have (i) if V' Ɔ V, with V = V(x), then V' = V(x); (ii) if V, V' = V(x), then VV' ¤ V(x).