Timeline for Subsets of $\mathbb{R}$, every nonempty subset of which generates a disconnected translation-invariant topology
Current License: CC BY-SA 4.0
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| Aug 6, 2021 at 18:23 | comment | added | Mike Krebs | Wow! Thanks for the rapid and spot-on answer. In case you or anyone else reading this thread is interested, your example shows that $\mathbb{R}$ does not possess a topology which is simultaneously connected, translation-invariant, and irresolvable. (The latter property means, a set and its complement cannot both be dense.) For if it does, then either $A$ or $A^c$ must have nonempty interior, but as you have shown, this will imply that $A$ is clopen. | |
| Aug 6, 2021 at 18:20 | vote | accept | Mike Krebs | ||
| Aug 4, 2021 at 10:30 | history | answered | KP Hart | CC BY-SA 4.0 |