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Jul
25
comment Good problem book on Abstract Algebra
Thanks Francis. The thread was useful. Fraleigh's book seems interesting. The solutions are exactly what I want. Since I am self-studying in isolation, it is useful to verify the solutions.
Jul
25
comment Good problem book on Abstract Algebra
I admit they are really good. Still, I believe it is better to look around just to see what problems do other people see around. I am not a math major and my college/university does not have these courses, so I am really scarce on exposure.
Jul
25
comment Good problem book on Abstract Algebra
Hi, thanks for your comments. Herstein does have a fresh perspective of problems and exposition. I read it and already like how he presented the three different proofs of Sylow Theorems including the very basic combinatorial one. Thanks
Jul
25
comment Continuous mappings pull back closed sets to closed sets
I was not taught that way, not using that or equivalent definition, I was taught as I wrote in that comment. While, I did learn the new definition, once in a while the old habit unconsciously takes over. I hope, I get used to the new one fast enough though :-)
Jul
24
asked Good problem book on Abstract Algebra
Jul
24
accepted Continuous mappings pull back closed sets to closed sets
Jul
24
revised Continuous mappings pull back closed sets to closed sets
added 207 characters in body
Jul
24
comment Continuous mappings pull back closed sets to closed sets
Okay, then it is trivial. You just take the complements and get done. Somehow, now, the question is something else, while the actual difficulty was something else. I guess, I would edit the question and someone can edit his answer and then, I would accept it so that it represents what actually happened?
Jul
24
comment Continuous mappings pull back closed sets to closed sets
I am very sorry, I meant to use $G'$, will correct it if possible.
Jul
24
comment Continuous mappings pull back closed sets to closed sets
I guess, I am not using the definition of $f^{-1}\left(G\right)$ properly. I am using the definition that $f^{-1}\left(G\right)$ exists only when $f$ is onto and if it is not then $f^{-1}\left(G\right)$ is a loose term for $f^{-1}\left(H\right)$ where $H$ is the range. If someone one can make it more clear to me, it would be better.
Jul
24
comment Continuous mappings pull back closed sets to closed sets
So lets say, $Z \in Y$ is the "range" of $f$ so that $f:X \rightarrow Z$ is an onto mapping. Then, if $G$ is closed in $Y$, then $G^{-1}$ is open in $Y$, but we are now considering $H = G^{-1} \bigcap Z$, how do I know that is also open? As far as I know, $H$ is open as a subset of $Z$ (which should be a subspace of $X$) in and only if it is the intersection with $Z$ of an open set which it is, but what about $H$ in $X$?
Jul
24
comment Continuous mappings pull back closed sets to closed sets
Definition of closed is that the a contains all its limit points where $a$ is a limit point of $X$ if for every open sphere $S_{\epsilon}\left(a\right)$ there exists an $x \in X$ such that $x \in S_{\epsilon}\left(a\right)$
Jul
24
comment Continuous mappings pull back closed sets to closed sets
Your second step would be true only for onto mappings, while the textbook says $f$ is an into mapping, or am I missing something?
Jul
24
revised Continuous mappings pull back closed sets to closed sets
Added the condition that the mapping is into.
Jul
24
comment Continuous mappings pull back closed sets to closed sets
But the mapping is an into mapping, how do I know $f^{-1}$ exists for all Y
Jul
24
asked Continuous mappings pull back closed sets to closed sets
Jul
21
revised How to prove that f is discontinuous at one point at least
deleted 475 characters in body
Jul
21
revised How to prove that f is discontinuous at one point at least
added 252 characters in body
Jul
21
answered How to prove that f is discontinuous at one point at least
May
13
awarded  Quorum