After I put out the first "Who cares about topology" video, someone pointed me to the paper that this video is based on as another great example of a topological fact being used to solve a non-topological problem. There's an interesting philosophical point to be had in the fact both this and the proof in the first topology video are non-constructive. That is, they show that some desired solution must exist, but not how to find it. This is true of just about every case I've seen of topology touching other fields.
This makes some sense, given that topology is all about understanding what properties are invariant under a wide variety of alterations you can make to some object, while specific solutions to specific instances of a problem tend to depend on the peculiarities of that instance. Nevertheless, I'd be curious to a fully constructive application of a pure topological fact, if that is not a contradiction.