This is an old math.SE question of mine that was never answered:
It is a nontrivial fact that there are only countably many isomorphism classes of compact Lie groups. One can prove this by a series of reductions: first to the connected case, then to the simply connected case, then by classifying simple Lie algebras. Of course, this proof actually gives a much stronger classification result.
If I only want to prove that there are countably many isomorphism classes of compact Lie groups, can I work without appealing to the classification of simple Lie algebras? I have some ideas involving Tannaka's theorem but I haven't worked out a proof yet.
The idea I had was to classify the possible symmetric monoidal [more adjectives if necessary] categories of representations of compact Lie groups; I think these categories are all "finitely presented" in a suitable sense, and from here it should be possible to show that there are only countably many presentations. Such presentations are given for the classical groups in Baez's Higher-Dimensional Algebra II: 2-Hilbert Spaces.