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Let's call a set "Pseudo compact" if it has the property that every closed cover (a cover consisting of closed sets) have a finite subcover.
Does "Pseudo Compact" in this case the same as "Anti-Compact" ? Then how can we describe the "Pseudo-Compact" subsets of Real Numbers?
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Your definition of “pseudo-compact” (every closed cover has a finite subcover) is actually the dual of compactness, commonly referred to as anti-compactness.
In ℝ, such sets must be finite because any infinite set can be covered by an infinite number of closed singletons, which do not have a finite subcover. Thus, only finite subsets of ℝ can be considered “pseudo-compact” in this regard.
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