Reputation
402
Next privilege 500 Rep.
Access review queues
Badges
2 12
Newest
 Yearling
Impact
~9k people reached

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 $<n$
When you quotient by $(X^n)$ you're effectively saying any time you see $X^n$ you can replace the term with 0. So all terms of degree $\geq n$ will vanish.
Feb
24
comment Any open subset of $\Bbb R$ is a at most countable union of disjoint open intervals. [Collecting Proofs]
What does "R is ccc" mean?
Feb
24
answered Any open subset of $\Bbb R$ is a at most countable union of disjoint open intervals. [Collecting Proofs]
Feb
19
awarded  Yearling
Feb
19
awarded  Scholar
Feb
19
comment Are countably infinite, compact, Hausdorff spaces necessarily second countable?
This is interesting. I didn't know that there was such a complete classification for these spaces. I'll have to do some reading up on the Sierpinski-Mazurkiewicz theorem. Thanks!
Feb
19
accepted Are countably infinite, compact, Hausdorff spaces necessarily second countable?