Timeline for Totally bounded spaces and axiom of choice
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Jan 16, 2018 at 19:05 | comment | added | Asaf Karagila♦ | @C.Eratosthene: The thing is that it is debatable whether or not the axiom of choice should be taken for granted (or at least countable choice), and only later on its uses should be pointed out, or if it should be explicitly debated from the get go. | |
| Jan 16, 2018 at 19:03 | vote | accept | C. Eratosthene | ||
| Jan 16, 2018 at 19:03 | comment | added | C. Eratosthene | @AsafKaragila: Thank you! Wikipedia article should probably be clarified because it gives this statement (every metric space that is complete and totally bounded is compact) as a fact before the discussion of the axiom of choice. Also, real analysis textbooks tend to emphasize the axiom of choice when it is used but I guess they take the axiom of countable choice for granted. | |
| Jan 16, 2018 at 18:06 | comment | added | Asaf Karagila♦ | @C.Eratosthene: Yes. There has been a lot of research into these sort of statements, and their equivalences (or lack thereof) to certain choice principles. Countable choice seems to be a central one in the case of metric and pseudometric spaces (and I'd imagine that generally in first-countable spaces too). | |
| Jan 16, 2018 at 18:06 | comment | added | C. Eratosthene | @AsafKaragila: Every metric space that is complete and totally bounded is compact. Does your example mean that this statement uses AC? | |
| Jan 16, 2018 at 17:52 | comment | added | Asaf Karagila♦ | @Mike: At least in the case I gave, nets are enough since the space is linearly ordered, every Dedekind cut corresponds to an obvious net. There is a lot of results about the equivalence between nets and filters, so I imagine the same can be said about filters. I will give it more thought. But in fairness, the completion of a metric space is defined as the Cauchy limits. It just shows there are two ways to complete. Just like there are two ways to define Noetherian without choice. | |
| Jan 16, 2018 at 17:40 | comment | added | Mike Shulman | Presumably this is only an issue with Cauchy sequences, and if you use more general nets or filters then the problem goes away? If so, I think it's misleading to present it as "without AC the completion of a totally bounded space might not be compact" -- I would instead say that "without AC it does not suffice to use sequences in defining the completion of a metric space, but one has to use nets or filters instead just as for a general uniform space". | |
| Jan 16, 2018 at 17:28 | history | answered | Asaf Karagila♦ | CC BY-SA 3.0 |