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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. 5) Definition: A Hausdorff Space is a topological space X with the property that for any pair of distinct elements x₁ and x2 in X, there are disjoint open sets U₁ and U2 with x₁ EU₁ and x2 € U2. Prove that every finite point set of a Hausdorff space is a closed set.

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. 5) Definition: A Hausdorff Space is a topological space X with the property that for any pair of distinct elements x₁ and x2 in X, there are disjoint open sets U₁ and U2 with x₁ EU₁ and x2 € U2. Prove that every finite point set of a Hausdorff space is a closed set.

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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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.
5) Definition: A Hausdorff Space is a topological space X with the property that for any pair of distinct elements x₁ and x2 in X, there are disjoint open sets U₁ and U2 with x₁ EU₁ and x2 € U2. Prove that every finite point set of a Hausdorff space is a closed set.
Transcribed Image Text:5) Definition: A Hausdorff Space is a topological space X with the property that for any pair of distinct elements x₁ and x2 in X, there are disjoint open sets U₁ and U2 with x₁ EU₁ and x2 € U2. Prove that every finite point set of a Hausdorff space is a closed set.
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