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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).

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).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Need part i and ii

which yields with ß = 1/α'. Page α′||x||2 ≤ ||x||1, for x ¤ E, ||x||2 ≤ ß||x||1, for x ¤ E, 5 5 of 20 4 Neighborhood Definition 4.1 (Neighborhood). Let X be a topological space. A neighborhood V c X of x € X is any 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 neighborhood of 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 have ZOOM +
Transcribed Image Text:which yields with ß = 1/α'. Page α′||x||2 ≤ ||x||1, for x ¤ E, ||x||2 ≤ ß||x||1, for x ¤ E, 5 5 of 20 4 Neighborhood Definition 4.1 (Neighborhood). Let X be a topological space. A neighborhood V c X of x € X is any 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 neighborhood of 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 have ZOOM +
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).
Transcribed Image Text: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).
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