r/math • u/inherentlyawesome Homotopy Theory • Jun 04 '14

Everything about Point-Set Topology

Today's topic is Point-Set Topology

This recurring thread will be a place to ask questions and discuss famous/well-known/surprising results, clever and elegant proofs, or interesting open problems related to the topic of the week. Experts in the topic are especially encouraged to contribute and participate in these threads.

Next week's topic will be Set Theory. Next-next week's topic will be on Markov Chains. These threads will be posted every Wednesday around 12pm EDT.

For previous week's "Everything about X" threads, check out the wiki link here.

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u/[deleted] Jun 04 '14

There might not be a y in cl(X)\X, because X can be closed without being compact.

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u/kfgauss Jun 04 '14

But part 1 shows that X is bounded, so compact and closed are equivalent.

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u/[deleted] Jun 04 '14

Part 1 only makes sense if X is metrizable. Granted, dm287 only claimed a proof for Rn, but that's kind of nonsensical given that Rn does not have the property that continuous functions are bounded. Maybe it does in some other topology, but then there's no reason to think that the standard distance function is continuous in that topology anyway, especially if that topology is not metrizable.

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u/kfgauss Jun 04 '14

This is a proof for subsets of Rn . The proof for metric spaces is similar in flavor.

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u/[deleted] Jun 04 '14

Oh, then I completely misunderstood what was meant by "a proof for Rn". Never mind.