alyafey22
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In the Principles of Mathematical analysis by Rudin we have the following theorem
If $$\mathbb{K}_{\alpha}$$ is a collection of compact subsets of a metric space $$X$$ such that the intersection of every finite sub collection of $$\mathbb{K}_{\alpha}$$ is nonempty , then $$\cap\, \mathbb{K}_{\alpha} $$ is nonempty .
If I understand correctly then this theorem states that if any finite intersection is nonempty then any arbitrarily intersection is also nonempty , right ?. I was trying to understand the proof but it wasn't so clear for me
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If $$\mathbb{K}_{\alpha}$$ is a collection of compact subsets of a metric space $$X$$ such that the intersection of every finite sub collection of $$\mathbb{K}_{\alpha}$$ is nonempty , then $$\cap\, \mathbb{K}_{\alpha} $$ is nonempty .
If I understand correctly then this theorem states that if any finite intersection is nonempty then any arbitrarily intersection is also nonempty , right ?. I was trying to understand the proof but it wasn't so clear for me
