Timeline for Connectedness in the language of path-connectedness
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 9, 2015 at 19:30 | comment | added | Terry Tao | While this construction may be too trivial to have much mathematical content, I think it may well have some metamathematical content, by helping to explain why many results concerning path-connectedness seem to "automatically" have analogues for topological connectedness (or vice versa). For instance, from this construction, one can adapt the proof of (say) the fact that the product of two path-connected spaces is path-connected, to conclude the same for topologically connected spaces (and note that the cardinality restriction is not a probem here). | |
| Mar 28, 2015 at 21:26 | comment | added | Goldstern | A more interesting follow-up question would be: Given $\kappa$, is there a not too large space (say: of weight $\kappa$, or perhaps cardinality or density $\kappa$, or perhaps we might allow $\kappa^+$ or $2^\kappa$?) which decides connectedness (in the way suggested by Dominic) for all small spaces (or weight or cardinality or density $\le \kappa$)? | |
| Mar 28, 2015 at 6:41 | comment | added | Włodzimierz Holsztyński | This $\ C\ $ works only for a given in advance family of connected spaces, and there is no any sharp upper bound on the size of $\ C.\ $ Thus I fail to see this construction as cool or very nice. On the contrary, it is routine, and it doesn't buy much. | |
| Mar 24, 2015 at 16:04 | comment | added | Dominic van der Zypen | Very nice, Joseph! | |
| Mar 24, 2015 at 15:32 | comment | added | jmc | Cool! This confirmatively answers the implicit follow-up after the initial negative answer. | |
| Mar 24, 2015 at 15:28 | history | answered | Joseph Van Name | CC BY-SA 3.0 |