Question

Let X be a topological space. Show that the following three statements are equivalent: (a) X is Noetherian. (b) Every open subset of X is compact. (c) Every subset of X is compact.

Name: let x be a topological space show that the following three statements are equivalent a x is noetherian b every open subset of x is compact c every subset of x is compact 89456
Uploaded: 2023-02-13T10:29:32-08:00
Duration: 4 min 36 s
Channel: Madhur L
Description: let x be a topological space show that the following three statements are equivalent a x is noetherian b every open subset of x is compact c every subset of x is compact 89456

          Let X be a topological space. Show that the following three statements are equivalent: (a) X is Noetherian. (b) Every open subset of X is compact. (c) Every subset of X is compact.

Added by Ismael W.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

02/13/2023

Step 1

Assume that X is Noetherian. Let U be an open subset of X. Since X is Noetherian, every open subset of X is Noetherian. Therefore, U is Noetherian. Since every Noetherian space is compact, U is compact. Thus, (a) implies (b). Show more…

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Let X be a topological space. Show that the following three statements are equivalent: (a) X is Noetherian. (b) Every open subset of X is compact. (c) Every subset of X is compact.