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May
20
comment Sieves and topology
It's amazing how clear this is now. Thanks. And about reading the book further: I can't wait :-)
May
20
comment Sieves and topology
Alright, I think I got the idea now. Thanks for all the help. @t.b.: if you want to post it as an answer I will accept it.
May
20
comment Sieves and topology
@SimonMarkett: Nope, sorry. This topic is very new for me.
May
20
comment Sieves and topology
@t.b.: Thanks; this makes perfect sense then. How about the first condition. This means that the first 'level' sets cover $X$?
May
20
asked Sieves and topology
May
18
revised Help me evaluate limit of sequence
edited tags
May
18
suggested rejected edit on Help me evaluate limit of sequence
May
18
comment Distribution Functions of Measures and Countable Sets
Thanks, I think I can see the point now. The only thing I have to believe is that modifying the bijection from $\mathbb{N}$ to $\mathbb{Q}$ to obtain a well-order in $\mathbb{Q}$ does not require any choice :-) But I guess this is a standard procedure; I'm quite a beginner in set theory.
May
18
comment Distribution Functions of Measures and Countable Sets
Yeah, I'm familiar with that construction. What I think is, that I'm missing something vital here, since I don't understand why this function would establish a well-order. It is clear that the resulting image of $\mathbb{N}$ is an order in $\mathbb{Q}$, but I don't know where the 'well' part comes from. I'll try to read Asaf's post again.
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.