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4) In the standard topology on R", a set is open if every point in the set has an e- neighborhood that is completely contained in the set. EU - x0 € € >0 s.t. {x: x = xo| < €} ≤ U - Show that this collection of sets is a topology in the sense of the definition. Definition: A topology on a set X is a collection T of subsets of X having the following properties. (1) Both the empty set and X itself are elements of T. (2) The union of an arbitrary collection of elements of T is an element of T. (3) The intersection of a finite number of elements of T is an element of T. A set X with a specified topology T is called a topological space. The subsets of X that are members of are called the open sets of the topological space.

4) In the standard topology on R", a set is open if every point in the set has an e- neighborhood that is completely contained in the set. EU - x0 € € >0 s.t. {x: x = xo| < €} ≤ U - Show that this collection of sets is a topology in the sense of the definition. Definition: A topology on a set X is a collection T of subsets of X having the following properties. (1) Both the empty set and X itself are elements of T. (2) The union of an arbitrary collection of elements of T is an element of T. (3) The intersection of a finite number of elements of T is an element of T. A set X with a specified topology T is called a topological space. The subsets of X that are members of are called the open sets of the topological space.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.3: Properties Of Composite Mappings (optional)
Problem 9E: Find mappings f,g and h of a set A into itself such that fg=hg and fh. Find mappings f,g and h of a...
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4) In the standard topology on R", a set is open if every point in the set has an e- neighborhood that is completely contained in the set. EU - x0 € € >0 s.t. {x: x = xo| < €} ≤ U - Show that this collection of sets is a topology in the sense of the definition.
Transcribed Image Text:4) In the standard topology on R", a set is open if every point in the set has an e- neighborhood that is completely contained in the set. EU - x0 € € >0 s.t. {x: x = xo| < €} ≤ U - Show that this collection of sets is a topology in the sense of the definition.
Definition: A topology on a set X is a collection T of subsets of X having the following properties. (1) Both the empty set and X itself are elements of T. (2) The union of an arbitrary collection of elements of T is an element of T. (3) The intersection of a finite number of elements of T is an element of T. A set X with a specified topology T is called a topological space. The subsets of X that are members of are called the open sets of the topological space.
Transcribed Image Text:Definition: A topology on a set X is a collection T of subsets of X having the following properties. (1) Both the empty set and X itself are elements of T. (2) The union of an arbitrary collection of elements of T is an element of T. (3) The intersection of a finite number of elements of T is an element of T. A set X with a specified topology T is called a topological space. The subsets of X that are members of are called the open sets of the topological space.
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