Timeline for Which spaces have enough curves
Current License: CC BY-SA 3.0
9 events
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| Jun 16, 2016 at 14:43 | comment | added | Pedro Lauridsen Ribeiro | @MartinSleziak I think that your first comment and js21's answer below (which is partly based on the former) go in a similar direction as Theorem 4.11 of Kriegl-Michor - after all, topological vector spaces are always locally path connected and the example I gave is sequential (as it should). Moreover, by parts (1) and (2) of the same Theorem, metrizable locally convex spaces (as in part (2) of js21's answer) and strong duals of Fréchet-Schwartz spaces also have the final topology with respect to continuous curves. This result also shows that first countability is not a necessary condition. | |
| Jun 16, 2016 at 13:18 | history | edited | David Spivak | CC BY-SA 3.0 |
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| Jun 16, 2016 at 10:20 | answer | added | Jeremy Brazas | timeline score: 7 | |
| Jun 16, 2016 at 9:14 | answer | added | js21 | timeline score: 8 | |
| Jun 16, 2016 at 7:36 | comment | added | Martin Sleziak | @PedroLauridsenRibeiro I see. It is quite an embarrassing mistake - if I wanted all functions to preserve curves, I had to take $Y$ indiscrete rather than discrete. Of course, the rest of the argument falls apart. So my second comment is completely irrelevant. (But I should probably still let it there, so that other users see what you are reacting to.) BTW the part you found in Kriegl's and Michor's book seem to be rather relevant for this question, at least if we restrict to the case that $X$ is locally convex topological vector space. | |
| Jun 16, 2016 at 5:51 | comment | added | Pedro Lauridsen Ribeiro | @MartinSleziak not necessarily. According to Theorem 4.11 (5), pp. 39-40 of the book of A. Kriegl and P.W. Michor, The Convenient Setting of Global Analysis (AMS, 1997), $X$ has the final topology with respect to continuous curves if $X$ is the strong dual of a Fréchet-Montel locally convex vector space. Such spaces are far from being discrete - take e.g. $X=\mathbb{R}^n$ or $X=$ the space of distributions of compact support on a second countable smooth manifold. | |
| Jun 16, 2016 at 5:14 | comment | added | Martin Sleziak | I am probably missing something, but it seems if we take $Y$ discrete then any function $f \colon X\to Y$ maps curves to curves (since any $c\colon I\to Y$ is continuous). Doesn't this mean that any space with enough curves must be discrete? (We take $Y$ to be the discrete space on the set $X$ and $f=id_X$.) | |
| Jun 16, 2016 at 5:01 | comment | added | Martin Sleziak | If I understand this correctly, this is the same as saying that $X$ has final topology w.r.t. the family of maps $C_X=\{ c\colon I\to X; c\text{ is continuous}\}$. I.e., $f\colon X\to Y$ is continuous if and only if $f\circ c$ is continuous for every $c\in C_X$. This also implies that $X$ is a quotient of disjoint sum of several copies of $I$ (one for each $c\in C_X$), therefore $X$ must be a sequential space. | |
| Jun 16, 2016 at 3:12 | history | asked | David Spivak | CC BY-SA 3.0 |