Bounded set which is not compact

You are asked to verify this in Exercise 4. You might think about what goes wrong if you try to apply the proof of Theorem 4. Take careful note of how the finite subcover property is used in the proof; the technique is common in proofs about compactness. But we will use an equivalent definition, which says that a set is bounded if there is a limit on how far apart two points in the set can be.

We can define the "diameter" of such a subset: Definition 4. Note that by this definition, the empty set is bounded, but we will not attempt to define the diameter of the empty set. We have shown that every compact subset of a metric space is closed and bounded.

Compact surfaces are more constrained. Non-compact ones can squirm out of your hands like blobs of rice pudding. Compact ones are more like jello: they might wobble a bit, but you can hold on to them if you don't mind getting your hands a little dirty. The post-rigorous understanding of compactness allows the word "compact" to circle around from something that feels like robot speak to something that aligns very closely with an English meaning of the word.

I like to think of it as a delightful accident of mathematical-linguistic convergence. The views expressed are those of the author s and are not necessarily those of Scientific American. Follow Evelyn Lamb on Twitter. Already a subscriber? Sign in. Thanks for reading Scientific American. Create your free account or Sign in to continue.

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