Timeline for Countable connected space where removing $1$ point destroys connectedness
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 30, 2018 at 18:01 | comment | added | მამუკა ჯიბლაძე | Actually one might argue that there are three different classes: empty (no pieces); connected (one piece); disconnected (more than one piece). | |
| Dec 29, 2018 at 18:11 | answer | added | KP Hart | timeline score: -1 | |
| Dec 26, 2018 at 8:54 | comment | added | Najib Idrissi | @FredRohrer Your statement is also true if $\varnothing$ is not connected. I think there are many reasons to want it to not be connected: unique decomposition in connected components; $\hom(X,-)$ preserves coproducts if $X$ is connected; for a (path-)connected space, $\pi_0(X) = *$; a product is connected iff both factors are connected. (This isn't off the top of my head, I'm reading this.) Anyway, this is a bit tangential, and as you say we're certainly rehashing old arguments... | |
| Dec 25, 2018 at 22:07 | comment | added | Fred Rohrer | @Najib: The (topological) reason that 1 is not prime is that the empty space is not irreducible. | |
| Dec 25, 2018 at 22:04 | comment | added | Fred Rohrer | @Najib: Surely this is a convention (or, rather, depends on the definition of connectedness). I think it is more important to be able to say that every point lies in a unique maximal connected subspace (i.e., its connected component). This is possible if the empty space is connected (which is by the way in accordance with Bourbaki). But this has been discussed on MO before... | |
| Dec 25, 2018 at 21:35 | comment | added | Thomas Rot | @FredRohrer: It depends on the convention indeed. | |
| Dec 25, 2018 at 21:32 | comment | added | Najib Idrissi | @FredRohrer This is a convention, but usually, I would say no. For the same reason that 1 isn't prime. Otherwise you cannot say that a space uniquely decomposes into a disjoint union of connected spaces... | |
| Dec 25, 2018 at 20:15 | comment | added | Fred Rohrer | @Thomas: But the empty space is connected! | |
| Dec 25, 2018 at 19:24 | comment | added | Thomas Rot | Can’t resist: The one point space satisfies this. | |
| Dec 25, 2018 at 18:17 | answer | added | Forever Mozart | timeline score: 4 | |
| Dec 25, 2018 at 17:10 | vote | accept | Dominic van der Zypen | ||
| Dec 25, 2018 at 10:38 | answer | added | user49822 | timeline score: 8 | |
| Dec 25, 2018 at 8:35 | history | asked | Dominic van der Zypen | CC BY-SA 4.0 |