Please do not rely too much on chatgpt, because its answer may be wrong. Please consider it carefully and give your own answer. You can borrow ideas from gpt, but please do not believe its answer.Very very grateful!Please do not rely too much on chatgpt, because its answer may be wrong. Please consider it carefully and give your own answer.You can borrowideas from gpt, but please do not believe its answer.Very very grateful! Given a set X ‡ Ø and p = X consider the collection of subsets ofTp = {U ≤X | p & U}U{X}.(i) Show that Tp is a topology on X. It is called the excluded point topology.(ii) Describe the closure {p} and the interior {p}° of the singleton set {p} with respectto the excluded point topology.(iii) Decide whether or not (X, Tp) is first countable and justify your answer.Exercise 1.X given byIn Lemma 2.15 we saw how taking the closure of a set interacts with the basic relationsand operations from set-theory (inclusions, unions, intersections). We can obviously ask thesame question about the operation of taking the interior of a set.

We have to show that τp is a topology on X, we need to verify three properties:1. Both the empty set…

Exercise 1. X given by Given a set X 0 and p = X consider the collection of subsets of Tp = {UCX|p&U}U{X}. (i) Show that Tp is a topology on X. It is called the excluded point topology. (ii) Describe the closure {p} and the interior {p} of the singleton set {p} with respect to the excluded point topology. (iii) Decide whether or not (X, Tp) is first countable and justify your answer.

Exercise 1. X given by Given a set X 0 and p = X consider the collection of subsets of Tp = {UCX|p&U}U{X}. (i) Show that Tp is a topology on X. It is called the excluded point topology. (ii) Describe the closure {p} and the interior {p} of the singleton set {p} with respect to the excluded point topology. (iii) Decide whether or not (X, Tp) is first countable and justify your answer.

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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