Topology - prove that X has a countable base

In summary, the conversation discusses finding a countable base for the topology of a topological compact space X. The solution involves proving X to be Hausdorff and then finding a collection of open subsets U_x\times V_x of X\times X such that for each n, (x,x)\in U_x\times V_x\subseteq G_n, where G_n is an open subset of X\times X. Taking all finite intersections of these subsets gives a base for the topology of X.
  • #1
rustyrake
5
0

Homework Statement



[itex]X[/itex] - topological compact space

[itex]\Delta = \{(x, y) \in X \times X: x=y \} \subset X \times X[/itex]

[itex]\Delta = \bigcap_{n=1}^{\infty} G_{n}[/itex], where [itex]G_{1}, G_{2}, ... \subset X \times X[/itex] are open subsets.

Show that the topology of [itex]X[/itex] has a countable base.

Homework Equations



The Attempt at a Solution



i have no idea what to start with. i don't really want to get a solution, just some clues...
 
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  • #2
Let's first find a candidate of a base, shall we??

For each [itex](x,x)\in G_n[/itex], we can find [itex](x,x)\in U_x\times V_x\subseteq G_n[/itex]. Now apply compactness on the [itex]U_x\times V_x[/itex].
 
  • #3
meaning: choose a finite number of them?
and compactness of what?
 
  • #4
rustyrake said:
meaning: choose a finite number of them?

Yes.

and compactness of what?

Of X.

You might also want to prove X to be Hausdorff...
 
  • #5
hm, ok, and i do this for each n, and get countable family of finite families of small open sets, and my base are all those small open sets?

micromass said:
You might also want to prove X to be Hausdorff...
what do you mean?
 
  • #6
rustyrake said:
hm, ok, and i do this for each n, and get countable family of finite families of small open sets, and my base are all those small open sets?

Not yet. You need to take all finite intersections as well.

Now try to prove that it is indeed a base. (you will need to make one last modification to the base in the end)

what do you mean?

Certainly you know what Hausdorff means?
 
  • #7
micromass said:
Not yet. You need to take all finite intersections as well.

Now try to prove that it is indeed a base. (you will need to make one last modification to the base in the end)

:( i don't get it. why intersections? and... intersections of what?


Certainly you know what Hausdorff means?
yes, but according to the definition of compactness i know, X is Hausdorff and i don't have to prove it.
 
  • #8
rustyrake said:
:( i don't get it. why intersections? and... intersections of what?

You found a collection of [itex]U\times V[/itex]'s. Now take all the finite intersections.

We will eventually want a base such that

[tex]\bigcap_{x\in G}{G}=\{x\}[/tex]

yes, but according to the definition of compactness i know, X is Hausdorff and i don't have to prove it.

Ah, ok. Never mind then.
 
  • #9
micromass said:
You found a collection of [itex]U\times V[/itex]'s. Now take all the finite intersections.

We will eventually want a base such that

[tex]\bigcap_{x\in G}{G}=\{x\}[/tex]

still don't get it... maybe it's too late. i'll think more about it in the morning.

but: if i take only [itex]U[/itex]'s from those [itex]U\times V[/itex]'s... why isn't it already our base?
 

1. What is topology?

Topology is a branch of mathematics that studies the properties of geometric figures that are unchanged under continuous transformations.

2. What is a countable base?

A countable base is a collection of open sets in a topological space that can be used to generate all other open sets in that space. It is called countable because the collection has a countable number of elements.

3. Why is it important to prove that X has a countable base?

Proving that X has a countable base is important because it allows us to understand the topological properties of X in a more manageable way. It also makes it easier to prove other topological theorems and results.

4. What does it mean for a topological space to have a countable base?

A topological space having a countable base means that it has a collection of open sets that is countable and that can be used to generate all other open sets in the space.

5. How do you prove that X has a countable base?

To prove that X has a countable base, we need to show that there exists a countable collection of open sets that can generate all other open sets in the space. This can be done by explicitly constructing such a collection or by showing that the space satisfies certain properties, such as being second countable.

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