Timeline for Is $\mathbb{R}\cong\text{Cont}(X,Y)$ for some non-trivial spaces $X,Y$?
Current License: CC BY-SA 3.0
11 events
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| Jan 8, 2018 at 19:57 | history | edited | Uri Bader | CC BY-SA 3.0 |
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| Jan 8, 2018 at 19:55 | vote | accept | Dominic van der Zypen | ||
| Jan 8, 2018 at 19:00 | comment | added | Uri Bader | @Taras, thanks. Indeed, if $Y$ is convex and $y\in Y$ is extreme, then the constant function $y\in C(X,Y)$ is extreme. Thus, $C(X,Y)-\{y\}$ is contractible. | |
| Jan 8, 2018 at 18:43 | comment | added | Uri Bader | @Todd, thanks for triggering me to edit. I initially felt uncomfortable with this question, so I was brief. In retrospect I see it gets upvoted, so suppose I was wrong to do so. | |
| Jan 8, 2018 at 17:55 | comment | added | Todd Trimble | Uri, thanks for the edit. I wasn't trying to nitpick -- I honestly did find the original hard to follow. Nice proof. | |
| Jan 8, 2018 at 17:52 | comment | added | Taras Banakh | Indeed, the proof of Uri Bader was correct (exploiting a very nice observation with the extreme point of $C(X,Y)$). Initially I did not catch the idea and because of that wrote my proof (actually a continuation of the proof of Uri Bader). | |
| Jan 8, 2018 at 17:05 | history | edited | Uri Bader | CC BY-SA 3.0 |
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| Jan 8, 2018 at 16:44 | comment | added | Uri Bader | @Todd, Third thing first, we didn't use anything about $X$ because we didn't need to. Second, $\mathbb{R}-\{1\}$ stands for $C(X,Y)$ minus the constant function 1, and for your first complain: that was my sloppy way to explain the obvious fact that $C(X,Y)-\{y\}$ is contactible where $Y$ is a convex set, $y\in Y$ is an extreme point and $y\in C(X,Y)$ represents the constant function. That seems obvious enough to me, so I allowed myself some informal presentation... | |
| Jan 8, 2018 at 15:06 | comment | added | Pietro Majer | In fact, that X has more than one point follows from the assumption Y is not homeomorphic to $\mathbb{R}=C(X,Y)$ | |
| Jan 8, 2018 at 14:36 | comment | added | Todd Trimble | Sorry to be dense, but I must admit that I'm not following the final sentence. First, I don't know how to parse postcomposing an object $C(X, Y)$ with a contraction map. Second, I'm not seeing where $\mathbb{R} - \{1\}$ is coming from. Third, we didn't use anything about $X$ having more than one point that I can see (maybe we don't actually need that condition?). | |
| Jan 8, 2018 at 13:54 | history | answered | Uri Bader | CC BY-SA 3.0 |