Actually, having an infinity at the end of every line on the complex plane just gives you a simple disc. To get the projective plane, you have to associate anti

Actually, having an infinity at the end of every line on the complex plane just gives you a simple disc. To get the projective plane, you have to associate antipodal points, like associating +/-infinity on the real line to a single infinity, but for every line that runs through the origin separately. But in topology, whether these points are 'at infinity' or not is arbitrary, because the complex plane is indistinguishable from the unit disc, until you start adding some more structure to it like a differential structure. And trust me, you do not want to be doing any sort of calculus on the complex plane with anything other than a single infinity. Conformal mappings really don't work well without that simplification. And with that simplification, everything is magical and you can integrate rainbows to get unicorns and candy canes. Nonetheless, your point is valid, it's possible to have more than a single point at infinity. However, regardless, it's more conventional to think of infinity as a particular point with some unusual properties, and in these cases with multiple infinities they're all basically the same. You don't have classes of infinities the same way you have classes of infiniteness.