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May
27
accepted Characterization of proper maps using filters
May
25
comment Characterization of proper maps using filters
@HennoBrandsma: It is not clear to me how I would use this to prove that $f$ is closed in the proof of the reverse implication: Pick a $y$ and an open $O$ such that $A := f^{-1}(\{y\}) \subseteq O$. Let $\mathcal U$ be an ultrafilter finer than $A$. Supposing $A$ is compact, $\mathcal U$ converges to some $x \in A$, hence $f[\mathcal U]$ converges to $y$. Since $\mathcal U$ contains $O$, we have that $f(O) \in f[\mathcal U]$, which means that $f(O)$ intersects every open set containing $y$. From this, how do I find a open $V$ such that $f^{-1}(V) \subseteq O$?
May
25
comment Characterization of proper maps using filters
Of course! Thanks.
May
25
comment Characterization of proper maps using filters
I don't see the contradiction. $f(U) \subseteq W$ implies that $U \subseteq f^{-1}(W)$ and I know that $f^{-1}(W)$ is open and contains $A$.
May
24
revised Characterization of proper maps using filters
added 779 characters in body
May
24
comment Characterization of proper maps using filters
Nevermind. I see that $\mathcal U_C$ is the trace of $\mathcal U$ on $C$. But it is not clear that the trace would exist to begin with.
May
24
comment Characterization of proper maps using filters
In the beginning, you use $A$ when I think you mean $C$. Also, it is not clear to me what $\mathcal U_C$ is.
May
24
comment Characterization of proper maps using filters
@PatrickDaSilva I believe I wanted to say that either $A$ or its complement is in $\mathcal U$. Sorry about that.
May
23
asked Characterization of proper maps using filters
May
23
accepted Dense implies strictly dense in TOP
May
23
comment Dense implies strictly dense in TOP
I was thinking about proving that $\overline{O \cap X} = \bar O$. Looks like my thoughts were in the right direction. Thanks.
May
23
revised Dense implies strictly dense in TOP
added 233 characters in body
May
23
asked Dense implies strictly dense in TOP
May
13
awarded  Caucus
Jan
29
accepted Filters and sequences
Jan
29
asked Filters and sequences
Dec
4
comment Probability density function of $\sigma X + \mu$.
Oops. Yeah, you guys are right. What I am wondering now is how you are dealing with the $\sigma < 0$ case.
Dec
4
comment How to prove these three norm equivalence problems
These are matrix norms: en.wikipedia.org/wiki/Matrix_norm
Dec
4
comment Probability density function of $\sigma X + \mu$.
For one thing, you expect that $\int_{\mathbb R} f_Z(x) \, dx = 1$, but if you use the formula for $f_Z(x)$ that you have, you don't get 1.
Dec
4
comment Poisson distribution and probability of random variables
I am not understanding your problem. You have the formulas you need: just replace $\lambda$ with 5 and simplify.