Timeline for Ordinal vs. cardinal dimension
Current License: CC BY-SA 3.0
19 events
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| May 1, 2018 at 17:28 | comment | added | Alec Rhea | @YemonChoi Those sound like exactly what I'm looking for -- if there's a better language choice than 'dimension' please feel free to edit. | |
| May 1, 2018 at 17:14 | comment | added | Yemon Choi | How liberally do you want to interpret "dimension"? There are various "ordinal indices" that occur in topology and in Banach space theory, e.g. Cantor-Bendixson rank or Szlenk index - would these be examples of the kind you're looking for? | |
| May 1, 2018 at 17:12 | history | edited | Michael Hardy | CC BY-SA 3.0 |
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| May 1, 2018 at 14:00 | comment | added | Alec Rhea | @JoelDavidHamkins That sounds fascinating, if you can find any references or remember any examples I would appreciate it greatly. (thank you for your current answer as well!) | |
| May 1, 2018 at 13:47 | comment | added | Joel David Hamkins | I think that there are some model-theoretic dimension concepts that give rise to ordinal dimensions. | |
| May 1, 2018 at 12:59 | answer | added | Gerald Edgar | timeline score: 7 | |
| May 1, 2018 at 0:09 | history | edited | Alec Rhea | CC BY-SA 3.0 |
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| May 1, 2018 at 0:00 | comment | added | Alec Rhea | @AndreasBlass Thank you for the clarification. | |
| May 1, 2018 at 0:00 | history | edited | Alec Rhea | CC BY-SA 3.0 |
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| Apr 30, 2018 at 23:52 | answer | added | Joel David Hamkins | timeline score: 2 | |
| Apr 30, 2018 at 23:49 | comment | added | Andreas Blass | Gro-Tsen's remark that $\mathbb R^X$ depends, up to isomorphism, only on the cardinality of $X$ is correct for all three of the product topology, the box topology, and the uniform topology. None of these refer to any order on $X$. | |
| Apr 30, 2018 at 22:57 | review | Close votes | |||
| May 2, 2018 at 7:31 | |||||
| Apr 30, 2018 at 22:47 | comment | added | Alec Rhea | @Gro-Tsen Are you thinking of the product topology, the box topology or the uniform topology? | |
| Apr 30, 2018 at 22:43 | history | edited | Alec Rhea | CC BY-SA 3.0 |
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| Apr 30, 2018 at 22:42 | comment | added | Alec Rhea | @Gro-Tsen The vagueness was intentional, I'll edit to make that clear (no pun intended). How about as ordered vector spaces? I'll have to think some more on the topological homeomorphism, that is a good point. | |
| Apr 30, 2018 at 22:39 | comment | added | Gro-Tsen | (contd.) For all definitions of "space" I can think of right now, $\mathbb{R}^X$ depends, up to isomorphism, only on the cardinality of $X$, so that $\mathbb{R}^\omega$ and $\mathbb{R}^{\omega+1}$ are isomorphic (e.g., homeomorphic as topological spaces), and it makes no sense to try to ascribe them different invariants. | |
| Apr 30, 2018 at 22:37 | comment | added | Gro-Tsen | It would help make your question less obscure if you clarified what you mean by "a space": a topological space? the spectrum of a commutative ring? something else? Or are you deliberately leaving the word "space" vague because you are interested in any kind of answer and it is, so to speak, part of the question? (The latter is legitimate, but, if so, you should still make it clear that you are deliberately making it vague. 😉) (contd.) | |
| Apr 30, 2018 at 22:32 | comment | added | Alec Rhea | Crossposted from MSE: math.stackexchange.com/questions/2757957/…. It received little attention for days, and the reference given in the comments is interesting but I would like a definition for spaces that do not inherently come with a topology. | |
| Apr 30, 2018 at 22:31 | history | asked | Alec Rhea | CC BY-SA 3.0 |