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## Main Question or Discussion Point

A subset K of a metric space X is said to be compact if every open cover of K contains a finite subcover.

Does not this imply that every open set is compact. Because let F is open, then

F= F [itex]\bigcup[/itex] ∅. Since F and ∅ are open , we obtained a finite subcover of F.

Am I missing something here?

Does not this imply that every open set is compact. Because let F is open, then

F= F [itex]\bigcup[/itex] ∅. Since F and ∅ are open , we obtained a finite subcover of F.

Am I missing something here?