Topology - prove that X has a countable base

[rustyrake]

Homework Statement

X - topological compact space

\Delta = \{(x, y) \in X \times X: x=y \} \subset X \times X

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

Show that the topology of X 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...

 

[micromass]

Let's first find a candidate of a base, shall we??

For each (x,x)\in G_n, we can find (x,x)\in U_x\times V_x\subseteq G_n. Now apply compactness on the U_x\times V_x.

 

[rustyrake]

meaning: choose a finite number of them?
and compactness of what?

 

[micromass]

meaning: choose a finite number of them?

Yes.

and compactness of what?

Of X.

You might also want to prove X to be Hausdorff...

 

[rustyrake]

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?

You might also want to prove X to be Hausdorff...

what do you mean?

 

[micromass]

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?

 

[rustyrake]

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.

 

[micromass]

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

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

We will eventually want a base such that

\bigcap_{x\in G}{G}=\{x\}

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.

 

[rustyrake]

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

We will eventually want a base such that

\bigcap_{x\in G}{G}=\{x\}

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

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