Timeline for Metrizability of topology of compact convergence
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 23, 2020 at 13:08 | vote | accept | ABIM | ||
| Mar 23, 2020 at 12:44 | vote | accept | ABIM | ||
| Mar 23, 2020 at 13:08 | |||||
| Mar 21, 2020 at 13:15 | answer | added | Henno Brandsma | timeline score: 2 | |
| Mar 19, 2020 at 17:51 | comment | added | ABIM | True, I guess there should be some type of topological regulairty like contractability? | |
| Mar 19, 2020 at 10:59 | comment | added | KP Hart | How does that work for the space of rationals where you take each $K_n$ a singleton (using some enumeration of $\mathbb{Q}$)? Don't you get the topolog of pointwise convergence that way? | |
| Mar 19, 2020 at 6:28 | comment | added | user131781 | This gives the metrisability condition in the second one. Here it is the completeness which is tricky. For this you need some version of the Kelley condition, i.e., that a function is continuous whenever its restriction to compacta is. I am not a point set topologist but flicking through my home library suggests that your conditions might not suffice. | |
| Mar 19, 2020 at 6:23 | comment | added | user131781 | There are two points here, one very elementary, one rather subtle. The first one involves the metrisability. This is more transparent in the following version. If a uniformity is defined by a sequence $(d_n)$ of pseudometrics, then it can be be specifies by a single one. The standard ploy is to use $\sum \frac 1{2^n}\frac {d_n}{1+d_ n}$. Separability plays no role. | |
| Mar 18, 2020 at 23:15 | comment | added | Pietro Majer | Also note that X can be isometrically embeded in a separable Banach space E, so that C(X,Y) is a subspace of C(Y,E), a Frėchet space. | |
| Mar 18, 2020 at 20:15 | comment | added | abx | Might be an overkill, but look at Bourbaki's General Topology X, §3.1, Corollary. | |
| Mar 18, 2020 at 18:55 | comment | added | ABIM | Haha, ya same here (not the biggest fan from working from home). Thanks for the tip, I made the modification :) | |
| Mar 18, 2020 at 18:54 | history | edited | ABIM | CC BY-SA 4.0 |
added 21 characters in body
|
| Mar 18, 2020 at 18:28 | comment | added | Jochen Wengenroth | Corona prevents me from checking the bookshelf in my office. Just note that the formula is not completely correct, if the metric $d_X$ is not bounded the series may diverge. You should replace $d_X$ by $\min\{d_X,1\}$. | |
| Mar 18, 2020 at 18:02 | history | edited | YCor | CC BY-SA 4.0 |
formatting, removed meta-info from title
|
| Mar 18, 2020 at 17:24 | history | asked | ABIM | CC BY-SA 4.0 |