It's not hard to do the former (which is intuitive), but I'm interested in a proof of the latter (which is un-intuitive at a glance, but can be pictured geometrically, and then translated into rigorous math).
@Chris'ssistheartist How did you start ? I'd be tempted to make some beta appear then do some substitution and play a bit with that, but I doubt it would give that huge result..
@BalarkaSen that was a mistake.. Didn't I prove everything else with quantifiers? Please do discuss pathconnectedness with me ... Not now but after few more days....
@Chris'ssistheartist Sure, I have plenty of motivation for my research, stemming from a desire to better understand the characters of certain representations of algebraic groups
@Chris'ssistheartist I study algebraic topology because it deals with algebraic invariants of spaces, i.e., it helps to determine which spaces are not homeomorphic by using simple, easy to compute algebraic objects - like groups, modules and rings.
@Chris'ssistheartist I study topology (and want to study algebraic topology.. an many more) because it is a subject (I feel) which talks about stuff not bounded by the normal geometry....
Although what I am studying now is an analogy between galois theory and covering spaces, hoping to get to use some topological facts and analogize it in galois theory to understand the absolute galois group better.
@Chris'ssistheartist we respect you. but it's not very nice to make fun of people who can't do your sums even though they're "high school stuff". just be nice :)
@Chris'ssistheartist The difference is that mine is a desire to understand something broader. Yours is a desire to to calculations. I have no problem with your desire, but it does not constitute mathematical research
@TobiasKildetoft Well, to do calculations, you sometimes need to do research, find new tools, new approaches, otherwise it were pretty difficult to do calculations. Maybe you might like to read, say, the first notebook of Ramanujan and then tell me again he didn't do research.
@Chris'ssistheartist I am not really that familiar with what he did, apart from his $\tau$ function. As far as I understand it, most of what he did was not really mathematics prior to being "discovered", due to the lack of any sort of proofs
@Chris'ssistheartist For example quotient spaces ..... They are so much filled with visualizations .... Stuff which we when children used to think were far apart are now being joined by a map... It so nice!!
Its just that your are too judgmental about all other parts of maths and I remember @MikeMiller telling me the first day "never be judgmental about maths"@Chris'ssistheartist
@Rememberme I'm convinced that mathematics is amazing in all its branches, but I attend the area I like the most.
I'm totally fascinated by the mathematical connections amongst integrals, series and limits, it's the most fascinating thing I ever attended.
I like it, I love it, it's like a burning passion to me, I wanna learn more about it and develop new tools, strategy to attend harder and harder integrals, series and limits (that have never be calculated so far or not even invented).
@Chris'ssistheartist Now this is wrong. You have made fun of, say, Balarka, on many an occasion for not being able to do one of your "elementary" problems.
You have said to me that I have never entered any of my country's top places with outstanding marks.... @Chris'ssistheartist No offense but isn't that humiliation ?
@BalarkaSen I don't see why explanations above would be better great than mine, that I also addressed to @TobiasKildetoft, I wanna learn more about this part of analysis and develop new tools, strategy to attend harder and harder integrals, series and limits. Besides that, I wanna develop tools for solving elementarily most of the classical problems about integrals, series and limits that usually require tough approaches to be solved.
I can even talk a lot about my stuff once I arrange my thoughts in English. I don't see anything more fascinating, more motivating than what I have, attend, but I respect what you do.
@Chris'ssistheartist But you never do talk about your stuff. You only talk about specific integrals and series. Whenever the talk turns towards you research, you don't want to say anything
So I got to prove that $\{A_\alpha\}\cup A$ is connected where $\{A_\alpha\}$ is the set of connected spaces of$X$ and $A$ is a connected subspace of $A$ such that $A\cap \{A_\alpha\}\neq \emptyset$
So I have though of a separation of $\{A_\alpha\}\cup A$ $Y=K\cup M$ where $K$ and $M$ are disjoint
every surjective map has a right inverse. but a SES might not have a section (which is a group hom) : an example is $\Bbb Z/2 \to \Bbb Z/4 \to \Bbb Z/2$
@SohamChowdhury Of course, in more generality, in categorical terms, "surjective" should be replaced by "epimorphism" which need not imply surjective, even in concrete categories
What I mean to say is that if I can show that for all $\alpha_i$ $A\cup A_{\alpha_{i}}$ where $A_{\alpha_{i}} \in \{A_\alpha\}$ is connected then will this imply what I wanted to show @Tobias
@Hippalectryon and finally my research reveals the closed form to $$\int_0^{\pi/2} e^{-\csc ^2(x)} \csc ^2(x) \left(\text{Chi}\left(\cot ^2(x)\right)+\text{Shi}\left(\cot ^2(x)\right)\right) \, dx$$
$\bigcup\{A_\alpha\}$ is in the separation $(K,M)$ then its either in K or in M . Now if $A_{\alpha_{i}}$ is in K then inductively $A_\alpha$ is in K . This shows M is nonempty...@Tobias
