show that any compact space is totally bounded
Question
show that any compact space is totally bounded
Solution
Sure, here is a step-by-step proof that any compact space is totally bounded:
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Let's start by defining our terms. A topological space X is said to be compact if every open cover of X has a finite subcover. A metric space (X, d) is said to be totally bounded if for every ε > 0, there exists a finite number of balls of radius ε that cover X.
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Assume X is a compact space. We want to show that X is totally bounded.
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Let ε > 0 be given. Consider the open cover of X consisting of all open balls of radius ε. Since X is compact, there exists a finite subcover of this open cover.
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This means there exists a finite number of balls of radius ε that cover X.
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Therefore, X is totally bounded.
This completes the proof.
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