Timeline for Can a connected planar compactum minus a point be totally disconnected?
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 27, 2010 at 20:57 | comment | added | Bill Johnson | Mine is maybe more elementary, Anton, but I like yours because it gives more information. Being an analyst, I also prefer path connectness, and I had not realized before reading your argument that you can easily use path connectedness to study spaces that are not path connected. I can see putting this to use on some problems on Lipschitz quotients. | |
| Feb 27, 2010 at 20:51 | vote | accept | HJRW | ||
| Feb 27, 2010 at 20:50 | comment | added | Anton Petrunin | Your argument is more natural (I simply do not feel comfortable with connected spaces --- path-connected are better). | |
| Feb 27, 2010 at 20:49 | comment | added | Bill Johnson | Yes, although by separability you can lower the logical strength a bit if you care about that. | |
| Feb 27, 2010 at 20:46 | comment | added | HJRW | I like this. So, Bill, just to be clear, you're invoking Zorn's Lemma to construct your minimal continuum? | |
| Feb 27, 2010 at 20:35 | comment | added | Bill Johnson | The intersection of a nested family of continua is again a continuum. You are right to ask about this--it is the only place that compactness is needed. | |
| Feb 27, 2010 at 20:19 | comment | added | LSpice | Oops, never mind, I see from above that it is a compact, connected metric space. | |
| Feb 27, 2010 at 20:12 | comment | added | LSpice | Bill, I'm sorry to look foolish: What is a continuum here? Is it a set of a certain cardinality, an image of an interval, a connected planar compactum, or something else? (I am no topologist, so I assume that the term is standard; I just haven't encountered it.) The “pass to a minimal sub-continuum” step worries me a little. | |
| Feb 27, 2010 at 20:04 | history | answered | Bill Johnson | CC BY-SA 2.5 |