Exploring Compactness and Closed Sets in Topology

[edcvfr]

Homework Statement

E is a compact set, F is a closed set. Prove that intersection of E and F is compact

Homework Equations

The Attempt at a Solution

On Hausdoff space (the most general space I can work this out), compact set is closed. So E is closed. So intersection of E and F is closed. That is a closed subset of compact set E so intersection of E and F is closed.

My problem is i can't generalize the proof to general topological space. If E is not in Hausdoff space, it is not necessarily closed. I don't know if I was going in the right direction or not. Please help me with it. Thank you very much.

 

[Dick]

Homework Statement

E is a compact set, F is a closed set. Prove that intersection of E and F is compact

Homework Equations

The Attempt at a Solution

On Hausdoff space (the most general space I can work this out), compact set is closed. So E is closed. So intersection of E and F is closed. That is a closed subset of compact set E so intersection of E and F is closed.

My problem is i can't generalize the proof to general topological space. If E is not in Hausdoff space, it is not necessarily closed. I don't know if I was going in the right direction or not. Please help me with it. Thank you very much.

Go straight to the definition. Pick any open cover of F. Can you think of a way of extending that to an open cover of E?

 

[edcvfr]

I thought about your idea before. I got this far:
F is closed. Call {G_i} class of sets such that F$\subset$$\bigcup$_iG_i and {H_i} class of sets such that E$\subset$$\bigcup$_iH_i. Because E is compact, H has finite number of subcovers. If we union G and H, we have open cover that covers both E, F and definitely E$\cap$F. However, because F is not compact (I read that a closed set is not necessarily compact, though it must be compact in Hausdoff space), G may have infinite number of subcovers. Union of G and H may have infinite subcovers so the intersection can't be proved to be compact.

Thanks for your help anyway.

 

[canis89]

I'm sorry, but, if you want to proof the compactness of the intersection between E and F, should you start from picking any open cover of that intersection, instead of F? That is, if you want to go straight by definition like Dick said.

Anyway, once you pick such open cover, try to augment it with some set such that the union, say M, equals(or contains, which implies equality) the topological space, say, X. From there, deduce that E is a subset of some open cover which is a subset of this M. Then, use the compactness of E until you complete the proof.

Sorry, if it's not clear. I'm currently writing it with my phone.

 

[edcvfr]

Sorry canis, I'm afraid I don't get your idea. Can you please elaborate on that? I apologize for my stupidity.

 

[canis89]

Alright, let me be more straightforward then. If you want to proof that the intersection of E and F, say, A, is compact by using the definition of compact set, then, you could start by picking any open cover of A. Then, (this is what I intend to say back then), union this open cover with the complement of A (which is the union between the complements of E and F). See if you can continue from there.

 

[edcvfr]

Alright, I tried to think by your hints and get some results. I'm not sure my explanation is correct. Please correct it for me:
Call G the finite set of open subcover of E, A' be an open cover of A. If we union A' and complement of F and call it M (open because F is closed), we still have an open cover of A. Moreover, M also cover E and can be reduced to a finite number of subcover because E is compact. After all M cover A and can be reduced so A is compact.
Sorry I can't figure out the meaning of union between A' and complement of E.

 

[canis89]

Yes, you're right. That's the idea. Forget about the union of A' and the complement of E. It's redundant. You only have to union A' with F complement. Now, can you write your argument more specific?

 

[edcvfr]

could it be more specific? To me it is comprehensive enough.

 

[canis89]

Oh, I just want to make sure that when you reduce M, the result must contain only the finite subfamily of A' (it must not contain F complement). But it is straightforward since $A=E\cap F$ cannot be a subset of F complement.

 

[edcvfr]

Thank you very much for helping me with this problem.