Recent content by yaa09d

Yes, it is clear. I was just confused why [tex]\left|\mathbb{Q}^{(B)}\right|=|\mathbb{Q}|\cdot|B|[/itex]. However, I found a proof for that on another forum, so I am ok with that now. Do you know if there is any textbook where that relation is proved? Is it a standard relation in algebra...

Thank you for your reply.

Thank you very much.

Thank you for the detailed reply. I thought if we accept CH, then 2^{\aleph_0}= \aleph_1 Can you recommend me a book to study the basics of cardinal numbers, please? I am a first year grad student. I am not familiar enough with cardinals.

I see. It's clear now. Thank you.

Thank you for your quick reply, but how is that clear?

Hey there! Is there any formula to determine the power of a cardinal number to another cardinal number? Thank you!

Let V be a vector space over an infinite field $\mathbf{k}$. Let \beta be a basis of V. In this case we can write V\cong \mathbf{k}^{\oplus \beta}:=\bigl\{ f\colon\beta\to \mathbf{k}\bigm| f(\mathbf{b})=\mathbf{0}\text{ for all but finitely many }\mathbf{b}\in\beta\bigr\}...

Hey there! I have the following question: Q: If we consider R and C as Q-vector spaces, then how can we show they are isomorphic? I know that if a two vector spaces have bases with the same cardinality, then they are isomorphic. Also, Zorn lemma tells us that every vector space has a...

That's a great idea! Thank you for the help.

Thank you jasomill! Actually, I understood the examples you mentioned on the real line. However I do not know how to write a formal proof for the statement. I tried the following : Let A={gi:ui----->R^n} be an atlas on M. Let f: B^n ----> B^n be a homeomorphism s.t. not differentiable at 0...

How do we prove this statement "the third problem in Lee's introduction to smooth manifolds. it says that given any topological manifold of dim > 0 with a smooth atlas, one can construct uncountably many distinct smooth structures." Thank you!