Timeline for Partially ordered set of compatible topologies
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 6, 2016 at 18:40 | vote | accept | Dominic van der Zypen | ||
| Sep 6, 2016 at 14:47 | comment | added | Will Brian | OK @DominicvanderZypen, I've done that now. | |
| Sep 6, 2016 at 14:46 | answer | added | Will Brian | timeline score: 1 | |
| Sep 3, 2016 at 7:56 | comment | added | Dominic van der Zypen | Thanks Will! That's already sufficient for me to see that there are no minimal elements as in the question in general. If you can copy your remark into an answer I'll accept it. - What I'm really keen to see is whether $\text{Cptb}(X,{\cal J})$ is a lattice, see my more recent question on that. | |
| Sep 2, 2016 at 14:01 | comment | added | Will Brian | Setting $\mathcal J = \{X\}$ makes $\mathrm{Cptb}(X,\mathcal J)$ just the lattice of topologies on $X$, and then the answer to your question is known to be negative. Do you want to put further restrictions on $\mathcal J$? | |
| Sep 2, 2016 at 13:55 | history | edited | Dominic van der Zypen | CC BY-SA 3.0 |
excluded a trivial case
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| Sep 2, 2016 at 13:48 | comment | added | Dominic van der Zypen | Thanks @TarasBanakh for your remark. I forgot to exclude $\tau = {\cal P}(X)$, the discrete topology. In the case that $X$ is finite, $\text{Cptb}(X,{\cal J})$ is finite, so if $\tau \neq {\cal P}(X)$, then $\tau$ is not the greatest element of the poset $\text{Cptb}(X,{\cal J})$, therefore there is a minimal element in $\text{Cptb}(X,{\cal J})$ above $\tau$. | |
| Sep 1, 2016 at 14:53 | comment | added | Taras Banakh | No, if $X$ is finite and the topology $\tau$ itself is minimal in $Cptb(X,\mathcal J)$. | |
| Aug 28, 2016 at 9:32 | history | asked | Dominic van der Zypen | CC BY-SA 3.0 |