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Jun
24
comment Density of linear span of idempotents in $L^{\infty}$
@DanielFischer Am I right that $f\in L^\infty$ is idempotent if and only if $f=\chi_S$ (characteristic function) for $S\subseteq \Omega$ measurable?
Jun
19
comment Coarsest and Finest Topology
Then it depends on whether the smallest generated topology is a subset of the given family or not. Maybe you could add the new assumption to the question and then someone more knowledgable than myself will write an answer.
Jun
19
comment Coarsest and Finest Topology
No. For example if the family does not contain the empty set.
Jun
18
comment Distribution of $\log X$
@t.b. There is a smoke signal for you. And a cave painting. Not sure you get pinged if I send you a comment but I'm going to try.
Jun
16
comment Another limit to evaluate: $ \lim\limits_{x \to \infty}\frac{x-\sin x }{x-\tan x}$
Copy pasting from the question into wolfram alpha gives $1$ for me.
Jun
14
comment “Every linear mapping on a finite dimensional space is continuous”
Isn't $T(E)$ of dimension $\le n$? (assuming $E$ is of dimension $n$)
Jun
13
comment In topology class, continuous and surjective problem
@topy It was a pleasure!
Jun
13
comment In topology class, continuous and surjective problem
@topy Regarding your question in the comment: If you endow $[a,b]$ with the subspace topology then $[a,y) = [a,b]\cap (-\infty,y)$ hence $[a,y)$ is open in $[a,b]$. It is not closed since if it was both closed and open we could write $[a,b]$ as a disjoint union of open sets which would contradict the fact that $[a,b]$ is connected.
Jun
13
comment In topology class, continuous and surjective problem
@topy They are open by the definition of the subspace topology: If $Y$ is any subset of $\mathbb R$ then a set $S\subseteq Y$ in the subspace topology on $Y$ is defined to be open if there exists an open set $O \subseteq \mathbb R$ such that $S= Y \cap O$. Now in your question $Y$ is $f([0,1])$. Since $(y,\infty)$ is open in $\mathbb R$ the set $f([0,1]) \cap (y,\infty)$ is open in $f([0,1])$ (in the subspace topology).
Jun
13
comment A problem about general topology.
@DanielFischer What I don't understand is: how can one check something is a base for the topology on $X$ when one is not given a topology on $X$?
Jun
7
comment Find the minimum distance that equal maximum inner product
Yes, I'm still trying to understand.
Jun
7
comment Find the minimum distance that equal maximum inner product
But left side of what?
Jun
7
comment Find the minimum distance that equal maximum inner product
Is there a condition missing or am I misunderstanding something?
Jun
7
comment Find the minimum distance that equal maximum inner product
Also, I'm not sure this holds. If $H=\mathbb R^2$ and $x$ is $(1,0)$, $M$ is the x-axis then the minimal distance between $x$ and the x-axis is zero but the minimal distance between points on the y-axis with $\|y\|=1$ is strictly greater than $0$.
Jun
7
comment Find the minimum distance that equal maximum inner product
What do you mean by "...LHS is $Px_0$ now how to show RHS is also $Px_0$"? I'm not being obtuse, I really don't understand. Could you elaborate a bit please?
Jun
7
comment How to prove uniform continuity problem!
@SwapnilTri Thank you for your kind words : )
Jun
6
comment Necessary conditions for $A=K+\operatorname{Ker}(\phi)$
@tomasz Yes but your first comment implies that my first comment does not pertain. Now I am thinking that it does: if you can apply the splitting lemma it is a possible answer to the question. No?
Jun
4
comment What does it mean to say that a forcing “collapses cardinals”?
@ArthurFischer Yes, that's what I meant. Thank you for your comment. : ) But I'm still confused how this implies that some cardinals are no longer cardinals in the extension.
Jun
4
comment What does it mean to say that a forcing “collapses cardinals”?
@AndresCaicedo I thought it meant that a cardinal will have lower cardinality in the extension.
Jun
4
comment Necessary conditions for $A=K+\operatorname{Ker}(\phi)$
Yes, I can prove it. That is: I don't think there is much to prove especially given your answer and your comment.