Question
Find, on the set of integers, a compact, Hausdorff topology.
Step 1
We need to find a topology on the set of integers, Z, that is both compact and Hausdorff. A topology is a collection of open sets that satisfies certain axioms. A space is compact if every open cover has a finite subcover, and it is Hausdorff if any two distinct Show more…
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The set [0,1] ⊂ ℝ is compact if ℝ is equipped with the usual topology. We see that 𝒲 = {(1/j,1-1/j)}_{j=2}^∞ ∪ {(-1/4,1/4), (5/6,7/6)} is an open cover of [0, 1]. Describe explicitly a finite subcover.
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