Timeline for First-countable topological monoids without local absorbing elements whose topology is induced by a semimetric
Current License: CC BY-SA 3.0
13 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Apr 23, 2014 at 14:22 | comment | added | Salvo Tringali | [...] About the origins of basic ideas in the area of asymmetric topology'' (in C. E. Aull and R. Lowen (eds.), Handbook of the History of General Topology, Vol. 3, Dordrecht: Kluwer (2001), 853-968), reports a letter by the same Fox where even a paracompact Hausdorff counterexample (to the $\gamma$-space conjecture) is provided. | |
| Apr 23, 2014 at 14:20 | comment | added | Salvo Tringali | [...] (see Distance function and the metrization problem, BAMS 43 (1937), 133-142). However, it is still true that not all first-countable topologies are semimetrizable, and I learned from R. Fox' work that the question is related to the $\gamma$-space problem (every semimetrizable space is a $\gamma$-space, and it took some time before a disproof of the converse). A Hausdorff counterexample is, in fact, given in: R. Fox, Solution of the $\gamma$-space problem, Proc. AMS 85 (1982), 606-608. And H.-P. A. Künzi, in his survey "Nonsymmetric distances and their associated topologies: [...] | |
| Apr 23, 2014 at 14:09 | comment | added | Salvo Tringali | Errata corrige. My "proof" that $(X,\tau)$ isn't semimetrizable when $X$ is countably infinite and $\tau$ is the cofinite topology on $X$ was flawed, and in fact, the contrary is true! This follows from a (straightforward) generalization of a theorem by W. A. Wilson dating back to the 1930s, which appears, e.g., as Theorem 6.3.50 in J. Goubault-Larrecq, Non-Hausdorff Topology and Domain Theory: Selected Topics in Point-Set Topology (New Math. Monographs 22, Cambridge Univ. Press, 2013), or can be recovered as an instance of a theorem by A. H. Frink on countably-based quasi-uniformities [...] | |
| Apr 20, 2014 at 20:43 | comment | added | Salvo Tringali | I'm considering monoids (to me, a monoid is a unital semigroup, or, if you prefer, a unital associative magma in the sense of Bourbaki). But I'm not sure to get the point of your question! What do you mean? Btw, my use of "in addition" in the present formulation of Q2 is misleading: every cancellative monoid is resilient. | |
| Apr 20, 2014 at 17:50 | comment | added | Włodzimierz Holsztyński | I don't see you mentioning $1$ (or $e$). Do you consider monoids or semigroups? | |
| Apr 20, 2014 at 17:21 | comment | added | Salvo Tringali | Sorry for the late reply (I was on holiday). In fact, there're 1st-countable topologies which are not semimetrizable (to be honest, I had no doubt about their existence, but it's only two days ago that I found a counterexample): this is the case, e.g., with the cofinite topology on a countably infinite set. So yes, I restated the OP to take into account your comments. | |
| Apr 20, 2014 at 17:21 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
Restated the question to take into account Eric and Chris' comments
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| Apr 16, 2014 at 19:53 | comment | added | Chris Schommer-Pries | Or take the free commutative monoid generated by X, also resilient, and it contains X as a proper component. | |
| Apr 16, 2014 at 14:22 | comment | added | Eric Wofsey | Also, you really want to assume that the topology is semimetrizable, not just that it is first-countable (I don't know whether these are equivalent, but I doubt they are). Chris's construction is quite flexible and can easily be modified to be "resilient". For instance, you could take $X\coprod\mathbb{N}$, where adding elements of $X$ to anything but $0$ is the same as adding $1$. | |
| Apr 16, 2014 at 14:19 | comment | added | Eric Wofsey | A monoid is resilient iff it has no idempotents besides the identity, since any idempotent $i$ is absorbing in the submonoid $\{1,i\}$ and any (locally) absorbing element is idempotent. | |
| Apr 16, 2014 at 14:03 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
added 67 characters in body
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| Apr 16, 2014 at 13:54 | history | asked | Salvo Tringali | CC BY-SA 3.0 |