152 Connectedness and Compactness Chap. 3 6. A space is totally disconnected if its only connected subsets are one-point sets. Show that a finite Hausdorff space is totally disconnected. malom then X is totally discons
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We are given a totally Hausdorff space, which means that for any two distinct points $x$ and $y$ in the space, there exist disjoint open sets $U$ and $V$ such that $x \in U$ and $y \in V$. Show more…
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2.36 Theorem If {Kα} is a collection of compact subsets of a metric space X such that the intersection of every finite subcollection of {Kα} is nonempty, then ∩ Kα is nonempty. Proof Fix a member K₁ of {Kα} and put Gα = Kαᶜ. Assume that no point of K₁ belongs to every Kα. Then the sets Gα form an open cover of K₁; and since K₁ is compact, there are finitely many indices α₁, . . . , αₙ such that K₁ ⊂ Gα₁ ∪ ··· ∪ Gαₙ. But this means that K₁ ∩ Kα₁ ∩ ··· ∩ Kαₙ is empty, in contradiction to our hypothesis. Corollary If {Kₙ} is a sequence of nonempty compact sets such that Kₙ ⊃ Kₙ₊₁ (n = 1, 2, 3, . . .), then ∩₁∞ Kₙ is not empty.
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