Apollo
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 Feb 4 comment Definition of topological space What do you mean by "arbitrary topology for a metric space"? Do you mean an arbitrary topology on the points of the metric space? If so, then there is no reason that it will necessarily coincide with the metric topology. The collection of all open subsets in the 'metric topology' will satisfy the general definition for a topology, but there are topologies which are non-metrizable. Thus the notion of a topological space is a generalization of metric space. Nov 23 awarded Yearling Nov 23 awarded Yearling Sep 30 awarded Explainer Sep 24 awarded Autobiographer Jun 27 comment Show $\lim_{n_\rightarrow \infty}\int_0^\infty ne^{-\frac{2n^2x^2}{x + 1}}dx = \infty$ (Oops, my comment had a silly mistake.) Your answer quite decisively settles the issue. Nov 23 awarded Yearling May 1 awarded Nice Answer Mar 28 comment Limit of the ratio of the logs, knowing the ratio I'd be surprised if you could say much about the relationship between the two ratios in general. Mar 28 comment Limit of the ratio of the logs, knowing the ratio Not necessarily: consider $f(x)=x^a$, $g(x)=x^b$ with $ab$ the first limit is $\infty$ but the second limit will be $a/b$ - this we can achieve any positive value for the second limit. If $f(x)=e^x$ and $g(x)=x$ then both limits are $\infty$. Nov 23 awarded Yearling Apr 27 answered Reference for general-topology Dec 27 awarded Nice Answer Nov 23 awarded Yearling Jul 1 awarded Necromancer Jul 1 awarded Revival Apr 18 comment On the equation $3a^2-4b^3=7^c$ BTW Twh article "Computing integral points on Mordellâ€™s elliptic curves" by Gebel, Petho, Zimmer shows that the torsion groups are trivial (Prop.3.1) and the article may give the tools to complete the proof of the result. Apr 1 answered A generic question on stochastic integrals Apr 1 answered On the equation $3a^2-4b^3=7^c$