Timeline for Embedding abelian cancellative Hausdorff topological semigroups into abelian Hausdorff topological groups
Current License: CC BY-SA 3.0
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| Apr 29, 2016 at 8:31 | history | edited | Alex Ravsky | CC BY-SA 3.0 |
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| Apr 29, 2016 at 8:24 | vote | accept | Salvo Tringali | ||
| Apr 29, 2016 at 8:14 | comment | added | Alex Ravsky | @SalvoTringali I posed your first question at our topological seminar seminar and Oleg Gutik suggested to consider the example which I added to my answer. | |
| Apr 29, 2016 at 8:14 | history | edited | Alex Ravsky | CC BY-SA 3.0 |
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| Jan 30, 2016 at 7:17 | comment | added | Alex Ravsky | @SalvoTringali Suppose that the additive semigroup $S$ of non-negative real numbers endowed with the topology described above is a subgroup of a topological group $G$. Then there exists a neighborhood $U$ of the common zero $0$ of the semigroup $S$ and the group $G$ such that $U\cap S=\{0\}$. Since $1-1=0$, the continuity the group operation on $G$ implies that there exists a neighborhood $V$ of the zero such that $(1+V)-1\subset U$. Then there exists a natural number $n$ such that $1/n\in V$. Then $1/n=(1+1/n)-1\subset ((1+V)-1)\cap S\subset U\cap S=\{0\}$, a contradiction. | |
| Jan 27, 2016 at 3:51 | comment | added | Salvo Tringali | Thank you for your answer, and sorry for the delay in replying (I had forgot about this thread...). The example with the Sorgenfrey line is very nice in its simplicity (I should have thought of it, as it's Example 1.2.1 in Arhangel’skii and Tkachenko's book on topological groups, here on my table), and the result you mention (about locally compact paratopological groups being topological groups) goes back to R. Ellis, I think. But what about the example with the non-negative real numbers? And what do the experts from your group say, if anything, about the 1st question? | |
| Nov 13, 2015 at 21:35 | history | answered | Alex Ravsky | CC BY-SA 3.0 |