Compact Sets: Need help understanding

In summary, my professor proved that if C subset of X is a compact and A subset of C is closed then A is compact.
  • #1
Buri
273
0
My professor proved the following:

If C subset of X is a compact and A subset of C is closed then A is compact.

Proof: Let U_alpha be an open cover of A.

A subset of X is closed implies that U_0 = X\A is open.

C is a subset of (U_0) U (U_alpha) and covers X.

In particular they cover C (possibly containing U_0).

Therefore, a finite subcover exists U_0, U_1,...,U_k of C and these also cover A.
Therefore, U_1,U_2,...U_k, cover A and hence, A is compact.

I can follow the proof, but don't really see the idea behind it all. I don't see why is it that you just can't get the finite sub cover from C and say it also covers A so its compact. I guess maybe an open cover of A might not be included in the collection of all open covers from C, so it won't be for ALL open covers there is a finite cover.

Furthermore, why do do we kick out the U_0 = X\A at the end? Is there a reason why we have to?

If someone could explain what I'm having problems with, or even go through the entire proof with explanations it would be even better.

Thanks!

EDIT:

Wait, is the reason why I have to get the X\A out is that I'm technically selecting ONE open cover when I add it in? And so I must show that X\A goes for sure, so I only keep the very general U_alpha?
 
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  • #2
No, if we have any open cover for A, then adding the open set, X\A, to it gives an open cover for C. Since C is compact, any open cover for it has a finite subcover. The reason you can "kick" X\A out at the end is that you are looking for a finite subcover of A and X\A contains no points of A. The reason you need to "kick" X\A out is that you do not know if it was in the original cover of A.
 
  • #3
Thanks a lot. I get it now, you've helped me before and I appreciate it. Thanks!
 

Related to Compact Sets: Need help understanding

What is a compact set?

A compact set is a mathematical concept used in topology that refers to a set of numbers that is "closed" and "bounded". This means that the set contains all of its limit points and is not infinitely large.

How do you determine if a set is compact?

There are a few different ways to determine if a set is compact, but one common method is to use the Heine-Borel theorem. This theorem states that a set is compact if and only if it is closed and bounded.

Can a set be compact in one space, but not in another?

Yes, a set can be compact in one topological space, but not in another. This is because the definition of compactness depends on the specific topological space and its properties.

What is the importance of compact sets in mathematics?

Compact sets are important in mathematics because they provide a way to analyze and understand the properties of a set in a topological space. They also have many applications in other areas of mathematics, such as in functional analysis and differential equations.

How are compact sets related to open and closed sets?

Compact sets are closely related to both open and closed sets. In fact, a set is compact if and only if it is closed and bounded. Additionally, open sets and closed sets can be used to construct compact sets through various operations such as unions and intersections.

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