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Jul
9
answered Relationship between groups that have the same group of homomorphisms to another group
Jul
9
comment Relationship between groups that have the same group of homomorphisms to another group
oh, sure we do, and it's actually a very interesting question. One can ask for the hom sets to be isomorphic as sets, or as groups. One can ask for the isomorphism to hold for more than just a single group $G$. One can ask for naturality conditions.
Jul
9
comment Relationship between groups that have the same group of homomorphisms to another group
what do you mean that the two hom sets are the same? If $A\ne B$, then the two hom sets are never the same.
Jul
9
answered Initial elements in Set and identity
Jul
9
comment Understanding definition of Riemann Integral
I suggest you go back and first fully understand the notion of limit of a function at a point. If you understand that concept, then this notion of integrability won't be so hard.
Jul
9
comment Understanding definition of Riemann Integral
In general you can't say anything about $L$. That is what integration techniques are for. It is possible though to characterise the Riemann integrable functions on a closed interval, i.e., giving a condition that guarantees the existence of an appropriate $L$. A function is Riemann integrable if, and only if, its set of discontinuities is countable.
Jul
9
answered Understanding definition of Riemann Integral
Jul
9
comment coproduct of lattices preserving filtered property of positive elements
In this construction then both meets and joins are computed component-wise. But then $(0,y)\wedge (x,0)=(0,0)$ for all $x\in L_1$ and $y\in L_2$. So if $z\in W$ is well above $0$, then either $z\ge (x,0)$ or $z\ge (0,y)$, with $x\succ 0$ and $y\succ 0$. But these are not closed under finite meets, so $W$ is not filtered.
Jul
9
comment Why do left and right switch when direction is reversed?
@ErickWong, yes, once that 180 degree turn is taken in order to trace your steps back. It is no longer a turn, but rather a reflection. Locally, you do the same, but globally things are very different.
Jul
9
answered Why do left and right switch when direction is reversed?
Jul
8
comment coproduct of lattices preserving filtered property of positive elements
@JonMarkPerry thanks, and corrected.
Jul
8
revised coproduct of lattices preserving filtered property of positive elements
edited body
Jul
7
answered If $G$ is abelian, it has a subgroup in every order of $|G|'s$ divisors?
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
30
answered Convergent $x_n,y_n$ and $x_n^{y_n}$ diverges
Jun
28
answered In the Hahn-Banach theorem, what is the purpose of the 'dominating function'?
Jun
28
asked coproduct of lattices preserving filtered property of positive elements
Jun
25
answered Is the Cauchy-Schwarz inequality ever used in Physics?
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.