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Jun
30
comment Convergent $x_n,y_n$ and $x_n^{y_n}$ diverges
To dispel any illusion that the sought counterexample is hard it should be noted that one can construct trivial counterexamples easily. While this example is pretty, it somewhat misses the point of the exercise.
Jun
30
comment Convergent $x_n,y_n$ and $x_n^{y_n}$ diverges
the problem is that the quantity $0^0$ is not well-defined. The book is fine.
Jun
28
comment Topological spaces with the same underlying set and basis.
Well, you need to pose a specific question. You must at least attempt a solution, explain where you got stuck, and then (perhaps) you'll get help here.
Jun
25
comment Prove that the image of a a closed and bounded interval in $\mathbb{R}$ is a a closed and bounded interval in $\mathbb{R}$?
You can't use compactness to prove the result. The image of a closed interval will be compact but there are lots of compact subsets of $\mathbb R $ which are not closed intervals.
Jun
23
comment Duality of Projective and Inductive Limit
In the sense that projective limit in $C^op$ is the same as injective limit in $C$. This is the meaning of duality in category theory.
Jun
23
comment Countable Connected Hausdorff Space
did you google "countable connected Hausdorff space"? You will get lots of hits.
Jun
23
comment $\mathbb{Q}(\sqrt2) $ is isomorphic to $\mathbb{Q}(\sqrt2 +2 )$
If they are equal, then they are isomorphic. You don't need any homomorphism (and I'm not sure which homomorphism you had in mind. The one you tried to describe is unclear.)
Jun
23
comment $\mathbb{Q}(\sqrt2) $ is isomorphic to $\mathbb{Q}(\sqrt2 +2 )$
They are equal, so yes, they are isomorphic.
Jun
21
comment How should one picture a topology/ topological space?
no, still not a topology and the last paragraph is still not good.
Jun
21
comment How should one picture a topology/ topological space?
the collection of all intervals in $\mathbb R$ is not a topology. You last paragraph is obscure. What do you mean by "take some arbitrary set, draw open balls lying in..." how can you draw open balls in an arbitrary set? what does that mean? The rest is just as unclear.
Jun
21
comment How should one picture a topology/ topological space?
Section 2 of arxiv.org/abs/1408.3887
Jun
16
comment Homotopy/fundamental group question: Why group axioms fail when defined on paths?
@ThomasAndrews of course, a monoid. Thanks (corrected). I changed the wording of the opening phrase to better explain the purpose of this answer.
Jun
16
comment Homotopy/fundamental group question: Why group axioms fail when defined on paths?
absolutely @ThomasAndrews I just thought it useful for OP to understand that it is not the desire to have a group that brings us to quotient by homotopy, it is the quotient that interests us. I remember as a student that the motivation for the fundamental group was presented as if we just really are looking for a group and follow our nose to do what it takes to obtain a group. This is misleading and the reason for my lengthy answer which, as you say, subsides OP's question, but, hopefully, places it in a different light.
Jun
10
comment a curve does not need to be injective?
yes @Mathcho, exactly.
Jun
9
comment Definition of a Cartesian Closed Category
@KevinCarlson you don't have to say what they are, but what OP did is not just not say what they are, but used a strange notation. I agree, this is probably just an interpretation of the wiki page stated, for whatever reason, using logical symbols. Obviously, if OP does not know what products and exponentials are, he should find out. That did not seem to be the question though.
Jun
9
comment Definition of a Cartesian Closed Category
en.wikipedia.org/wiki/Cartesian_closed_category
Jun
9
comment Definition of a Cartesian Closed Category
you can find the conventional notation in plenty of sources.
Jun
9
comment Version of the Axiom of Induction for Real Induction?
It's not clear what your question is. Of course you can state the principal of real induction (as appearing in the article you quote) using logical symbols and notation, much like you can do with any well-formed notion in mathematics.
Jun
9
comment Version of the Axiom of Induction for Real Induction?
I'm not sure what else you want then. The article you quote proves that what is called there the principal of real induction is equivalent to the completeness property (it even goes further to prove the same for general ordered sets). It is a restatement of some standard proof techniques in $\mathbb R$ and other posets.
Jun
9
comment Word in English to mean “study the convergence / divergence” of a series?
"study the nature of the series" is a perfectly valid English sentence, conveying what you explain (though it is not limited just to convergence issues. Just like the French word implies, it can also mean to study if the series is converging fast or slow, if it is positive or alternating, etc.).