Timeline for In what rigorous sense are Sperner's Lemma and the Brouwer Fixed Point Theorem equivalent?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| May 30, 2013 at 13:06 | vote | accept | James Propp | ||
| May 30, 2013 at 13:06 | vote | accept | James Propp | ||
| May 30, 2013 at 13:06 | |||||
| May 23, 2013 at 16:20 | comment | added | François G. Dorais | I doubt it. These weaker systems try to capture complexity theory rather than computability theory, they are inherently finite in nature and so incorporating compactness doesn't make much sense. Maybe there is another way, but I don't know any. Compactness is pretty strong so other restrictions are likely to be lost in its shadow. | |
| May 23, 2013 at 14:39 | comment | added | Timothy Chow | François, your parenthetical comment is very close to what I was hoping for. My only remaining question is then whether some compactness principle can be adjoined to one of those "weaker base systems" to produce a rigorous formulation of "Sperner = Brouwer". | |
| May 22, 2013 at 15:13 | comment | added | François G. Dorais | Tim, isn't the Approximate Fixed Point Theorem exactly the "interesting part" of Brouwer's? (The Approximate Fixed Point Theorem is equivalent to Sperner's Lemma over some much weaker base systems but I don't think that's published anywhere.) | |
| May 22, 2013 at 14:22 | comment | added | Timothy Chow | François's answer is of course correct; however, it is precisely this example (Sperner's lemma vs. Brouwer's fixed point theorem) that has often made me wonder if there's an alternative to SOSOA that more closely matches our intuition about the equivalence between the two results. The "mathematician in the street" might be inclined to toss some amount of compactness into the background theory since it has the feel of a general logical principle. Somehow the "interesting" part of Sperner is the same as the "interesting" part of Brouwer, and it would be cool if we could formalize this intuition. | |
| May 22, 2013 at 8:19 | comment | added | domotorp | For someone not into axioms, this can be the proof that they are practically "equivalent" - only one more line (compactness) is needed to derive Brouwer from Sperner. | |
| May 22, 2013 at 5:37 | history | answered | François G. Dorais | CC BY-SA 3.0 |