Timeline for Can you explicitly write $\mathbb{R}^2$ as a disjoint union of two totally path disconnected sets?
Current License: CC BY-SA 2.5
6 events
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| Feb 26, 2015 at 5:09 | comment | added | Włodzimierz Holsztyński | This answer above, as I've pointed out in a previous comment, is not correct. The correct one is given in a new thread/question which has a similar title but without word "path". | |
| Mar 13, 2013 at 13:22 | comment | added | Gerald Edgar | "the union of two zero dimensional sets has dimension at most 1" ... Decomposition Theorem $\dim X \le n$ if and only if $X$ can be written as a union of $n+1$ sets, each of dimension $\le 0$. Engleking, Dimension Theory (somewhere on pages 257 to 260). | |
| Mar 13, 2013 at 7:25 | comment | added | Greg Martin | Remedial question: what's a reference for the statement "the union of two zero dimensional sets has dimension at most 1"? | |
| Feb 24, 2013 at 20:42 | comment | added | Włodzimierz Holsztyński | @Gerald: What is your definition of "totally disconnected"? A subset of square which does not contain any connected subsets but the empty set or 1-element sets, can still have topological dimension = 1. Together with a 0-dimensional set they can cover the whole square. | |
| Nov 20, 2010 at 21:14 | comment | added | George Lowther | @Gerald: The two sets in my construction are connected and locally connected, yet totally path disconnected. This came up in a recent MO question (mathoverflow.net/questions/46748/…). | |
| Oct 31, 2010 at 13:28 | history | answered | Gerald Edgar | CC BY-SA 2.5 |