Compactness of the Classical Groups

There’s also the special versions of these groups, which have the additional requirement that each element has determinant 1.

We’ll look at the differences between the real and complex groups, in particular if they are compact or not. I’m using the nonstandard characterization of compactness because it made it really easy to figure out the answer in my head for each case. Which is why I’m writing this.

Definition: A set \(S\) in a topological space \(X\) (\(S \subseteq X\)) is compact if and only if every point \(s\) in its nonstandard extension \(^*S\) is nearstandard (infinitely close to some standard point \(p\) in the original set)1.

So \(\mathbb{R}\) is not compact because its nonstandard extension is \(\mathbb{R}^*\), which includes infinite numbers. The open interval \((0, 1)\) is not compact because its nonstandard extension is \((0, 1)^*\) which includes infinitesimal numbers like \(\varepsilon\), which is infinitely close to 0, except that 0 is missing because it’s an OPEN interval.

This is the part about compactness generalizing “closed and bounded” in Euclidean space. In fact, it even explains why the “closed and bounded” definition works in the first place. The only nonstandard elements in Euclidean space that aren’t nearstandard are infinite numbers, so as long as the set is closed (contains all its limit points), then boundedness guarantees compactness.

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