Christian Bueno
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 Apr 25 comment Is directed set countable, if for each element there are only finitely many smaller ones? Thought I'd share an observation that clarified things for me: After some thought, it occurred to me that a directed set is countable if and only if every element has countably many predecessors and there exists at least one element with countably many successors. From here, it's clear that every counter-example to my original question involves a directed set with all elements having uncountably many successors (of which the above is a great example). Apr 25 awarded Yearling Apr 25 accepted Is directed set countable, if for each element there are only finitely many smaller ones? Apr 22 awarded Nice Question Apr 21 comment Is directed set countable, if for each element there are only finitely many smaller ones? Neither. The two would be incomparable. However, (2,4) is larger than both for example. Apr 21 asked Is directed set countable, if for each element there are only finitely many smaller ones? Jan 26 comment Expected Number of Loops from $n$ Ropes In case this helps the reader: the probability of $\frac{1}{2n-1}$ comes from having $\binom{2n}{2}$ possible pairs to tie together, only $n$ of which lead to a loop (selecting both ends from the same rope). Hence the probability of making a loop is $\frac{n}{\binom{2n}{2}}=\frac{n}{(2n)(2n-1)/2}=\frac{1}{2n-1}$. Jan 26 comment Using Stirling's Approximation to Find Maximum I think there must be an error in your question. The series the you have written down has no maximum since it is just the exponential function $e^x$ which has no max. en.wikipedia.org/wiki/… Oct 28 comment Dense simple smooth immersed curves in manifolds. @JackLee Yes. My understanding is that "simple curve" and "injective curve" mean the same thing. Unless "simplicity" does not rule out self-tangency. Oct 27 asked Dense simple smooth immersed curves in manifolds. May 20 comment Intuition behind independence & conditional probability This is a great explanation. And it works where conditional probability often is undefined (i.e. when $P(B)=0$). Mar 9 comment What is the predual of $L^1$ I think $K$ also needs to be Hausdorff no? Feb 25 comment Any open subset of $\Bbb R$ is a at most countable union of disjoint open intervals. [Collecting Proofs] Ok thanks for clearing that up. Feb 24 comment If $R=K[X]/(X^n)$, can represent any element as polynomial with degree \$