Timeline for Connectedness of deleted symmetric product
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 17 at 20:27 | comment | added | Peluso | @GregoryArone You are right. In the Hausdorff case, which is the one that interests me, they are homeomorphic. Thank you so much. | |
| Mar 16 at 17:40 | comment | added | Gregory Arone | Have a look at this question and answer mathoverflow.net/questions/293942/… . I think the answer more or less tells you that they are homeomorphic, at least when $X$ is a Hausdorff space. | |
| Mar 16 at 15:53 | comment | added | Peluso | @GregoryArone I don't have a proof available, but I think it's not the same topological space, is it? The space $[X]^n$ and the space $C_n(X)$ are not homeomorphic. | |
| Mar 16 at 14:34 | comment | added | Andy Putman | @AlessandroCodenotti: To start with, not all open sets are of that form. | |
| Mar 16 at 10:42 | history | edited | Gregory Arone |
edited tags
|
|
| Mar 16 at 9:42 | comment | added | Alessandro Codenotti | Maybe I'm missing something, but doesn't the following argument work? Let $U=\langle U_0,\ldots, U_n\rangle\subseteq [X]^n$ be an open set which is nonempty and not the whole space. We show it is not closed. By assumption there must be $i$ such that $U_i\neq X$, and since $X$ is connected, $U_i$ is not closed. Assume wlog $i=0$ Let $x_\lambda\to x$ be a net in $U_0$ converging to a point $x$ not in $U_0$. Let $u_j\in U_j\setminus U_0$, for $j>0$ be arbitrary. The net $E_\lambda=\{x_\lambda,u_1,\ldots,u_n\}$ converges to $\{x,u_1,\ldots,u_n\}$ which is not in $U$. | |
| Mar 16 at 7:49 | comment | added | Gregory Arone | If I am reading your question correctly, it is a duplicate of this one math.stackexchange.com/q/148868 , which has some informative answers. | |
| Mar 16 at 4:32 | history | asked | Peluso | CC BY-SA 4.0 |