Thomas E.
Reputation
4,761
Next privilege 5,000 Rep.
Approve tag wiki edits
 May 18 comment Distribution Functions of Measures and Countable Sets Thanks for clearing it up. I have never seen the explicit formula for the bijection, you know where I could find it? And how do you know that this function well-orders the rationals? May 18 comment Distribution Functions of Measures and Countable Sets What do you mean by avoiding axiom of choice and yet choosing a well-ordering of $\mathbb{Q}$? Isn't well-ordering equivalent with axiom of choice? May 17 comment Product space and product topology The notion $\Pi$ is just to shorten the product symboling. If I understood correctly what you mean, then they are not only isomorphic (did you mean homeomorphic?) but identical. I.e. $\Pi_{i\in I}X_{i}=\Pi_{i=1}^{2}X_{i}=X_{1}\times X_{2}$ (where $I=\{1,2\}$). Just a matter of notation. May 17 comment Which of the following define a metric on $\mathbb{R}$? Note that $d_{1}(-1,1)=0$. May 17 comment Axiom of choice and compactness. @AsafKaragila: I have Thomas Jech's Set Theory, so that's probably what I will start with since it's accessible for me. Thanks alot for your time and input. May 17 comment Axiom of choice and compactness. @AsafKaragila: Thanks. What would you suggest? May 17 comment Axiom of choice and compactness. Alright, thanks a lot Asaf. I have both books in my hands now, can't wait to study them later today. My knowledge of set theory is not in a very high level, but I'm getting there slowly :-) May 16 comment Axiom of choice and compactness. Thanks; that link contains some nice discussions. May 16 accepted Axiom of choice and compactness. May 16 comment Axiom of choice and compactness. Thanks for the excellent reply. I was aware of compactness being equiv with ultrafilter compactness in ZFC, but I will look up Herrlich's material for sure. The infinite Dedekind-finite set sounds very interesting. What do you think would be the best source to look it up from? May 16 asked Axiom of choice and compactness. May 16 revised Dealing with Tychonoff's Theorem. deleted 4 characters in body May 16 answered Dealing with Tychonoff's Theorem. May 16 comment Dealing with Tychonoff's Theorem. What do you mean by Heine-Borel definition of compactness? May 16 answered Cantor set: Lebesgue measure and uncountability May 16 comment A metric space problem By choosing $A=B$ we see that it does not imply. Did you mean instead that does it imply $A\cap B\neq \emptyset$? May 16 revised Connection of ideas of Measuring to the Measure theory . added 27 characters in body May 16 answered Connection of ideas of Measuring to the Measure theory . May 16 comment What does a simple function actually mean? By the way, a simple function is continuous iff $\partial A_{i}=\emptyset$ for all $i$, so they can represent other continuous functions than constants depending on the underlying topology. In $\mathbb{R}^{n}$ and Euclidean topology you're right though, since the only sets without boundary are $\mathbb{R}^{n}$ itself and $\emptyset$. May 16 answered What does a simple function actually mean?