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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.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 13E: 13. Consider the set of all nonempty subsets of . Determine whether the given relation on is...
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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 borrow
ideas 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 of Tp = {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 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 In Lemma 2.15 we saw how taking the closure of a set interacts with the basic relations and operations from set-theory (inclusions, unions, intersections). We can obviously ask the same question about the operation of taking the interior of a set.
Transcribed Image Text:Given a set X ‡ Ø and p = X consider the collection of subsets of Tp = {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 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 In Lemma 2.15 we saw how taking the closure of a set interacts with the basic relations and operations from set-theory (inclusions, unions, intersections). We can obviously ask the same question about the operation of taking the interior of a set.
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