Given any compact metric space $(M, d)$ (which in the article is the closed unit square), the set of non-empty compact subsets of $M$ under the Hausdorff metric $h$ is also a compact metric space.
Yeah, seems like Turing defines a symbol to be a compact subset of the unit square. Whether that makes sense or not to you is up to interpretation I guess, but like you say the closure should be more or less the same as the symbol for "reasonable" symbols.
You then need to assume that each symbol has some variation in how it's written, I think, which you can think of as an open set in the symbol-space (which is compact under $h$ as per my previous comment). The collection of all these opens forms a cover of the symbol-space, which by compactness must have a finite sub-cover (Turing's "alphabet").
