A1 Let X be a topological space (or metric space, if you prefer). Prove that the following conditions are equivalent. (i) X is compact. (ii) If F = {Fa}oej is any collection of closed subsets of X that satisfies the finite intersection property, then F = Fa + Ø. Hint: Just proceed using the definitions: bear in mind that open sets and closed sets are com- plements of one another; bear in mind also that unions (present in the definition of open covers and compactness), intersections (present in condition (ii) above), and complements are related via DeMorgan's Laws, which we used in class on various occasions.
A1 Let X be a topological space (or metric space, if you prefer). Prove that the following conditions are equivalent. (i) X is compact. (ii) If F = {Fa}oej is any collection of closed subsets of X that satisfies the finite intersection property, then F = Fa + Ø. Hint: Just proceed using the definitions: bear in mind that open sets and closed sets are com- plements of one another; bear in mind also that unions (present in the definition of open covers and compactness), intersections (present in condition (ii) above), and complements are related via DeMorgan's Laws, which we used in class on various occasions.
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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![A1 Let X be a topological space (or metric space, if you prefer). Prove that the following
conditions are equivalent.
(i) X is compact.
(ii) If F = {Fa}oej is any collection of closed subsets of X that satisfies the finite intersection
property, then NF = N Fa + Ø.
Hint: Just proceed using the definitions: bear in mind that open sets and closed sets are com-
plements of one another; bear in mind also that unions (present in the definition of open covers
and compactness), intersections (present in condition (ii) above), and complements are related via
DeMorgan's Laws, which we used in class on various occasions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fceb54a1e-6a60-4744-9db1-8508c0c7ded7%2F20960f35-8bc5-4f70-ae6b-ecfe6755688e%2Ftq0963i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A1 Let X be a topological space (or metric space, if you prefer). Prove that the following
conditions are equivalent.
(i) X is compact.
(ii) If F = {Fa}oej is any collection of closed subsets of X that satisfies the finite intersection
property, then NF = N Fa + Ø.
Hint: Just proceed using the definitions: bear in mind that open sets and closed sets are com-
plements of one another; bear in mind also that unions (present in the definition of open covers
and compactness), intersections (present in condition (ii) above), and complements are related via
DeMorgan's Laws, which we used in class on various occasions.
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