Recent content by mcastillo356

Have a nice Thanksgiving Day!

Do you mean "Id est"? What software are you using? Anyhow, at PF it is not needed LaTeX to type ie; but in case you needed, it should be four #, and ie in the middle; ie, ##ie##.. Hope it helps.

You were there, I remember you answering to my doubts I keep you in my heart.

AI is not reviewed. I mean it doesn't mention sources of the given information.

I study by myself. It's tough. The thread explains it perfectly.

Lemma 5 and Lemma 6 seem easy, but I've had to think about set partitions and equivalence relations. Attempt For any equivalence relation on a set ##X##, the set of its equivalence classes is a partition of ##X##. Conversely, from any partition ##P## of ##X##, we can define an equivalence...

Lemma 2 is difficult to me Why ##A=A_{0}\sim{B_{1}}\sim{A_{2}}\sim{B_{3}}\sim{A_{4}}\sim{\cdots}##? Attempt ##A=A_{0}##, ##B_{1}=f(A_{0})##, and so on. Why is this a consequence of injectivity? Attempt A set mapping is injective if...

Working hard on it.

"The sting", that's my favorite movie. The soundtrack is amazing.

##\mathbb{R}^+##?

No idea. I can only attempt at it: ##F## could be the identity function? @martinbn says so. Another doubt: The Axiom of Choice is needed?

I'm working on the Schroether-Bernstein theorem, this is, the next step in the textbook "Introduction to topology and modern analysis", by Simmons. We assume that ##f:\,X\rightarrow{Y}## is a one-to-one mapping of ##X## into ##Y##, and that ##g:\,Y\rightarrow{X}## is a one-to-one mapping of...

Yes. Simmons is being precise. Into means inyective. That's accurate, indeed. Thanks, @fresh_42 !

The textbook is being fine. I asked the forum for some introduction to topology, and decided to start with Simmon`s. This naive question is due to ignorance of the words into and onto, which I don't distinguish in Spanish. A quick browsing sugests I'm right.

In fact, the cartesian coordinates are continous, and sine function too; this means the value of the limit at any points ##(x,\,y)## will be the same as the function's value. In this case: $$\displaystyle\lim_{(x,\,y)\rightarrow{(a,\,b)}}{f(x,\,y)g(x,\,y)}=LM$$ ##L## and ##M## being the values...