So, here I am again.(adsbygoogle = window.adsbygoogle || []).push({});

This is to only have a proof reviewed. Like I said before, I will do this from time to time, so I know I'm staying on track.

Theorem - Cantor Theorem

Let [itex]{C_1, C_2, ...}[/itex] be a monotone decreasing countable family of non-void closed subsets of a T_1 space such that C_1 is countably compact. Then the intersection of the family above is non-void.

Proof

Now, create the subspace C_1 with respect to the T_1 space. Call this subspace C.

Since C_1 is countably compact the subspace C is countably compact, and it is also a T_1 space (herediatary).

Now, by Theorem 1.3 (proven), which says that...

Theorem 1.3

Let X be a countably compact T_1 space, and let F be a countable collection of open subsets of X. If F covers X, then some finite subfamily of F covers X (X is compact).

So, replacing X with C we get C is compact.

Now, also by Theorem 1.10 (proven), which says that...

Theorem 1.10

A space X is compact if and only if each family of closed subsets of X with the FIP (finite intersection property) has a non-null intersection.

So, now [itex]{C_1, C_2, ...}[/itex] form a family of closed subsets in C. Also, since they are non-void it is quickly observed that they have the FIP.

So, since C is compact this implies that the intersection of [itex]{C_1, C_2, ...}[/itex] is non-empty.

And we are done.

The early proofs made it very easy of course. This is what makes the text so good in my opinion. Proving small little things leads to an easy proof of something bigger.

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