Timeline for Partitioning $\beta \mathbb{Z} \setminus \mathbb{Z}$
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 10, 2019 at 15:12 | vote | accept | André Caldas | ||
| Dec 10, 2019 at 13:35 | comment | added | Will Brian | Yes, that's right. If $f: \mathbb N^* \rightarrow X$ is constant on sets of the form $p+\beta \mathbb Z$, then its image is connected. Furthermore, as a sort of converse, if $X$ is a connected space of weight $\leq\! \aleph_1$, then there is a continuous surjection $\mathbb N^* \rightarrow X$ that is constant on sets of the form $p+\beta \mathbb Z$. | |
| Dec 10, 2019 at 10:02 | comment | added | André Caldas | Also, I concluded that if I have a continuous function from $\mathbb{N}^*$ that is constant over the sets $p + \beta \mathbb{Z}$, then, its image should be connected. Right? | |
| Dec 9, 2019 at 23:55 | comment | added | André Caldas | You made lots of things clear to me! I have learned a lot! I found out that I really wanted to partition $\mathbb{N}^*$ using sets of the kind $p + \beta \mathbb{Z}$. Fortunately, you answered a little more then I asked, and showed this wonderful $R$ set. I understood that every minimal right $P$-ideal [I just invented this name :-)] is one equivalence class, and those sets are very abundant per your work with Verner. | |
| Dec 9, 2019 at 23:06 | comment | added | Will Brian | You might not be missing anything. They do not need to be open or closed -- they just happen to be so in both of the examples I give. As you say, one could come up with a partition into neither-open-nor-closed sets by taking unions of the kinds of right ideals I describe. Probably there are other ways to do it too. The last paragraph (the one in parentheses) is just meant to show that there's only one non-trivial partition into clopen sets that works -- is this what's not clear? | |
| Dec 9, 2019 at 22:41 | comment | added | André Caldas | I might be missing something... but I do not understand why the sets on the partition need to be open or closed. I understand $p + \beta \mathbb{Z}$ is closed, but the elements on the partition could be made from an infinite union of them. | |
| Dec 9, 2019 at 18:25 | history | answered | Will Brian | CC BY-SA 4.0 |