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 Jul 25 comment Good problem book on Abstract Algebra I am now reading the Berkeley Problems in Mathematics and it seems good. Lets see how well can I solve the problems now. 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