Timeline for Which topological spaces contain dense simply connected subspace?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 4, 2017 at 14:18 | comment | added | erz | @JeremyBrazas why cannot we just remove a point from each of the circles? | |
| May 4, 2017 at 8:11 | answer | added | Francesco Polizzi | timeline score: 3 | |
| May 3, 2017 at 17:59 | answer | added | Will Brian | timeline score: 3 | |
| May 3, 2017 at 17:50 | comment | added | Pietro Majer | @erz: any non-empty open subset of the Sierpinski carpet $S$ contains the boundary of some square $Q$ disjoint from $S$ (one of the squares removed in the construction). Of course $\partial Q$ is a loop which is not contractible in $\mathbb{R}^2\setminus Q$, and a fortiori in $S$ . | |
| May 3, 2017 at 17:22 | comment | added | Christian Remling | The infinite-dimensional torus $S^{\mathbb N}$ with product topology is a more obvious example perhaps of a space with no simply connected open sets. | |
| May 3, 2017 at 16:01 | comment | added | erz | @PietroMajer Although it looks very intuitive, how exactly do you prove it? | |
| May 3, 2017 at 15:52 | comment | added | Pietro Majer | A negative example for the stronger property is the Sierpinski carpet en.wikipedia.org/wiki/Sierpinski_carpet , a connected compact plane set, whose non-empty open subsets are not simply connected. | |
| May 3, 2017 at 14:26 | history | asked | erz | CC BY-SA 3.0 |