I just started learning topology and I don't understand this example. The infinite union of all finite subsets of N (members of T4) would surely produce a set of all natural numbers? Wouldn't this set belong to T4 (N)? Why is it not?
It doesn't contain one, but it's infinite.
So it isn't N, it isn't a finite subset, and it isn't the empty set.
>>8067007
Its missing {1} so it cant be N
Just like if you only picked even numbers you'd have a subset not in the topology
This is a perfect question for /sqt/...
Topology without Tears I see
>>8067009
Could you clarify your first sentence? Why isn't the infinite union of all finite subsets of N just N itself?
>>8067016
>Why isn't the infinite union of all finite subsets of N just N itself?
Can you read? It's the union of all one point subsets except for {1}.
>>8067012
Samefag, back with a helpful response:
This is not a topology on \mathbb{N} since the set of open sets must be closed under unions. Certainly, the singleton sets are open. But if you union all the even-numbered singleton sets, you end up with an infinite subset of \mathbb{N}, which is not a member of \tau_4. Thus, \tau_4 cannot be a topology.
>>8067024
Could you quickly explain the idea of open and closed sets in this context? Thanks.
this infinite union doesnt contain 1, so it clearly isnt N. however it is an infinite subset of N, but T4 consists by definition only of finite subsets of N, therefore this union is not element of t4, so T4 cant be a topology snce the axiom of closedness under infinite unions isnt fulfilled
>>8067030
Certainly. A topological space is a set X together with a subset of the power set of X, which we will denote \tau_X.
The subsets of X which are members of \tau_X are called "open". Furthermore, some properties must be satisfied if X is indeed a topological space:
1. \emptyset \in \tau_X
2. X \in \tau_X
3. The union of any (possibly infinite) collection of subsets in \tau_X must also be in \tau_X.
4. The intersection of any finite collection of subsets in \tau_X must also be in \tau_X.
When I say "closed under unions", I'm saying axiom #3 must hold.
Lastly, a subset Y of X is called "closed" if its complement X \setminus Y is open.
>>8067043
Thanks that explains a lot. On a seperate note, how can I turn "\tau_X" and "\setminus" into actual mathematical symbols? I assume there is a browser extension for it but I can't seem to find it..
>>8067064
You use 4chan's terrible compiler. At the top left of the box that you type your reply to, there's this thing that says "TeX". So if I were to make that anon's post better looking you'd have:
Certainly. A topological space is a set [math]X[/math] together with a subset of the power set of [math]X[/math], which we will denote [math]\tau_X[/math].
The subsets of [math]X[/math] which are members of [math]\tau_X[/math] are called "open". Furthermore, some properties must be satisfied if [math]X[/math] is indeed a topological space:
1. [math]\emptyset \in \tau_X[/math]
2. [math]X \in \tau_X[/math]
3. The union of any (possibly infinite) collection of subsets in [math]\tau_X[/math] must also be in [math]\tau_X[/math].
4. The intersection of any finite collection of subsets in [math]\tau_X[/math] must also be in [math]\tau_X[/math].
When I say "closed under unions", I'm saying axiom #3 must hold.
Lastly, a subset [math]Y[/math] of [math]X[/math] is called "closed" if its complement [math]X \setminus Y[/math] is open.
Attached a screenshot of what I typed into the post to make it look nicer.
>>8067064
Sorry; I'm lazy, and I haven't tried to figure out how to get LaTeX to render on /sci/. See pic related though; I fired up my TeX environment and compiled it.
>>8067074
Though personally I'd use my preferred definition of a topological space:
Let [math]X[/math] be some set.
A collection [math]\mathcal{T}[/math] of subsets of [math]X[/math] (i.e. [math]\mathcal{T} \subset \mathcal{P}(X)[/math]) is called a topology if:
(i) [math]X, \varnothing \in \mathcal{T}[/math]
(ii) if [math]X_1, \ldots , X_k \in \mathcal{T}[/math], then [math]X_1 \cap \ldots \cap X_k \in \mathcal{T}[/math]
(iii) if [math]\{X_i : i \in \mathcal{A}\}[/math] is any collection of elements of [math]\mathcal{T}[/math], then [math]\displaystyle\bigcup_{i \in \mathcal{A}} X_i \in \mathcal{T}[/math]
>>8067082
Not picking at your post or anything, rather just some possible stylistic advice:
You may want to use (in plaintext) \tau_{_X} instead to get a better subscript, and \varnothing for a nicer empty set.
Also, for a nicer \tau, you can use the upgreek package so you can use \uptau instead.
>>8067090
Thanks for the formatting advice. Very helpful!
the fuck OP, it clearly says
{2} u {3} u ...
and you ask about the union of all finite subsets.
You were so enthusiastic to make a new thread, you didn't even bother reading the page another time.
Feel bad for fucking up this board. (You're amongst a lot of other bad people though, so I can understand if you don't end up feeling all that bad about it..)
thx
