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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?