2) Definition: Let X be a topological space. A closed set is the complement of an open set (CX-U for an open set U). Prove the following statements. (1) Both the empty set and X itself are closed sets. (2) The intersection of an arbitrary collection of closed sets is closed. (3) The union of a finite number of closed sets is closed. 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.
2) Definition: Let X be a topological space. A closed set is the complement of an open set (CX-U for an open set U). Prove the following statements. (1) Both the empty set and X itself are closed sets. (2) The intersection of an arbitrary collection of closed sets is closed. (3) The union of a finite number of closed sets is closed. 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
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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Transcribed Image Text:2) Definition: Let X be a topological space. A closed set is the complement of an open
set (CX-U for an open set U). Prove the following statements.
(1) Both the empty set and X itself are closed sets.
(2) The intersection of an arbitrary collection of closed sets is closed.
(3) The union of a finite number of closed sets is closed.
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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