Theorem from Baby Rudin (Thm. 2.36)

**Mollier** · January 10th 2011, 11:16 PM

Theorem. If $\{K_{\alpha}\}$ is a collection of compact subsets of a metric space $X$ such that the intersection of every finite subcollection of $\{K_{\alpha}\}$ is nonempty, then
$\bigcup_{\alpha}K_{\alpha}$ is nonempty.

I try and prove theorems my self before reading them (seldom looks pretty), and the one I came up with for this theorem is the following.

Theorem 2.34 says that compact subsets of metric spaces are closed. From this I know that every set in $\{K_{\alpha}\}$ is closed.

Theorem 2.24(b) says that for any collection $\{K_{\alpha}\}$ of closed sets, $\bigcap_{\alpha}K_{\alpha}$ is closed.

Now comes my question. Since the empty set is both closed and open, can I just say that $\bigcap_{\alpha}K_{\alpha}$ is nonempty since it is closed? I am pretty sure the answer is no, but it would be nice if someone could explain this a bit.

Thanks.

**Drexel28** · January 10th 2011, 11:44 PM

Originally Posted by Mollier

Theorem. If $\{K_{\alpha}\}$ is a collection of compact subsets of a metric space $X$ such that the intersection of every finite subcollection of $\{K_{\alpha}\}$ is nonempty, then
$\bigcup_{\alpha}K_{\alpha}$ is nonempty.

I try and prove theorems my self before reading them (seldom looks pretty), and the one I came up with for this theorem is the following.

Theorem 2.34 says that compact subsets of metric spaces are closed. From this I know that every set in $\{K_{\alpha}\}$ is closed.

Theorem 2.24(b) says that for any collection $\{K_{\alpha}\}$ of closed sets, $\bigcap_{\alpha}K_{\alpha}$ is closed.

Now comes my question. Since the empty set is both closed and open, can I just say that $\bigcap_{\alpha}K_{\alpha}$ is nonempty since it is closed? I am pretty sure the answer is no, but it would be nice if someone could explain this a bit.

Thanks.

Ok, let me see if I can get a handle on this. One knows that compact subspaces of metric spaces are closed-true. Next, one knows that the arbitrary intersection of closed sets is closed-true. So your argument is that by these two theorems one would have that $\displaystyle \bigcap_{\alpha\in\mathcal{A}}K_\alpha$ is a closed subspace of $X$ and thus can't be empty since $\varnothing$ is both open and closed? Unfortunately this is not the case :|. It's often true that a subspace of a metric space is both open and closed (clopen) for it not to be in general the space would have to be connected. It's often impossible (in full generality) to complete arguments such as this by looking at openess/closedness (although it works sometimes, like proving things like the fundamental lemma of path connectedness in alg. top.).

For the argument I'll try to give you two well placed hints:

Hint 1: If $\displaystyle \bigcap_{\alpha\in\mathcal{A}}K_\alpha=\varnothing$ then $\displaystyle \bigcup_{\alpha\in\mathcal{A}}\left(X-K_\alpha\right)=X$ .

Hint 2: If for every finite $\mathcal{B}\subseteq\mathcal{A}$ one has that $\displaystyle \bigcap_{\beta\in\mathcal{B}}K_\beta\ne\varnothing$ then for every finite $\mathcal{B}\subseteq\mathcal{A}$ one has $\displaystyle \bigcup_{\beta\in\mathcal{B}}\left(X-K_\beta\right)\subsetneq X$ .

So, since the $K_\alpha$ are closed we know that the $X-K_\alpha$ are open and so the above shows we've found a ____________ with no _____________.

**Jose27** · January 11th 2011, 06:43 PM

In fact, it's easy to see from Drexel28's argument that your property (which I've heard being called "finite intersection property") is equivalent to compactness (even in arbitrary topological spaces).

**Mollier** · January 13th 2011, 12:48 AM

Originally Posted by Drexel28

Ok, let me see if I can get a handle on this. One knows that compact subspaces of metric spaces are closed-true. Next, one knows that the arbitrary intersection of closed sets is closed-true. So your argument is that by these two theorems one would have that $\displaystyle \bigcap_{\alpha\in\mathcal{A}}K_\alpha$ is a closed subspace of $X$ and thus can't be empty since $\varnothing$ is both open and closed? Unfortunately this is not the case :|. It's often true that a subspace of a metric space is both open and closed (clopen) for it not to be in general the space would have to be connected. It's often impossible (in full generality) to complete arguments such as this by looking at openess/closedness (although it works sometimes, like proving things like the fundamental lemma of path connectedness in alg. top.).

For the argument I'll try to give you two well placed hints:

Hint 1: If $\displaystyle \bigcap_{\alpha\in\mathcal{A}}K_\alpha=\varnothing$ then $\displaystyle \bigcup_{\alpha\in\mathcal{A}}\left(X-K_\alpha\right)=X$ .

Hint 2: If for every finite $\mathcal{B}\subseteq\mathcal{A}$ one has that $\displaystyle \bigcap_{\beta\in\mathcal{B}}K_\beta\ne\varnothing$ then for every finite $\mathcal{B}\subseteq\mathcal{A}$ one has $\displaystyle \bigcup_{\beta\in\mathcal{B}}\left(X-K_\beta\right)\subsetneq X$ .

So, since the $K_\alpha$ are closed we know that the $X-K_\alpha$ are open and so the above shows we've found a __member__ with no __brain__.

Sorry for the late reply.
I understand the proof in the book to some extent, but I fail to see how your argument works.
If you have the time, I would appreciate it if you explain some more.
Thanks.

**Drexel28** · January 13th 2011, 12:58 AM

Originally Posted by Mollier

Sorry for the late reply.
I understand the proof in the book to some extent, but I fail to see how your argument works.
If you have the time, I would appreciate it if you explain some more.
Thanks.

Suppose that $\displaystyle \bigcap_{\alpha\in\mathcal{A}}K_\alpha=\varnothing$ then by DeMorgan's law we get that $\displaystyle \bigcup_{\alpha\in\mathcal{A}}\left(X-K_\alpha\right)=X$ but since the $K_\alpha$ are closed we see that $X-K_\alpha$ are open and so by compactness there exists some finite subcover $X-K_{\alpha_1},\cdots,X-K_{\alpha_n}$ but $\displaystyle \bigcup_{j=1}^{n}\left(X-K_{\alpha_j}\right)=X$ implies, once again by DeMorgan's law, that $\displaystyle \bigcap_{j=1}^{n}K_{\alpha_j}=\varnothing$ contradicting the FIP of $\left\{K_{\alpha}\right\}_{\alpha\in\mathcal{A}}$ .

**Mollier** · January 13th 2011, 08:45 PM

Thanks Drexel, much appreciated.

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