Timeline for Inserting an open and simply-connected set between a compact set and an open set
Current License: CC BY-SA 3.0
8 events
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| Feb 13, 2015 at 23:58 | comment | added | fedja | If the complement of $K$ is connected, just consider the set of points of the complement to which you can drag an open disk of fixed very small radius from infinity without touching $K$. It is a closed set and its complement $V$ is open, contains $K$, and is simply connected just because to drag the center of the disk inside a loop, you need to drag it to the boundary of the loop first (though you will need some kind of induction argument to show it with full rigor and completely from scratch). | |
| Feb 12, 2015 at 3:10 | comment | added | Paul Fabel |
Riemann mapping is nuclear weapons given the relatively crude inequalities of the original questions. Perhaps the original inquiry helps amplify a trichotomy among true'',obvious'', and ``why exactly is this true?''.
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| Feb 11, 2015 at 21:41 | comment | added | Mathieu Baillif | Thanks, Paul. That's what I had in mind, but the Riemann mapping theorem is in my mind heavy artillery, that's why I thought there was an "elementary" argument that I had not seen. | |
| Feb 11, 2015 at 21:07 | comment | added | Paul Fabel | If we allow the Riemann mapping theorem, take a conformal map from the (punctured) open unit disk minus (0,0) onto the complement of K. The images in the plane of the circles centered at (0,0) of radius 1/n will do. | |
| Feb 11, 2015 at 19:26 | comment | added | Mathieu Baillif | I admit that I don't see how to easily choose the curves so that the intersection of their `interiors' is $K$ without some artillery. | |
| Feb 11, 2015 at 15:17 | history | edited | Paul Fabel | CC BY-SA 3.0 |
added 3 characters in body
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| Feb 11, 2015 at 15:07 | comment | added | Allen Hatcher | Interesting use of language here (twice) that I don't recall seeing before: "If [X], if [Y], then [Z]." Is this equivalent to "If [X] and [Y], then [Z]"? Or maybe "If [X] such that [Y], then [Z]". | |
| Feb 11, 2015 at 14:34 | history | answered | Paul Fabel | CC BY-SA 3.0 |