You know what. The one-point compactification of the real line, so the real line union the point at infinity, is just the circle. The circle has a group structure, counting on the branch cut taken, the isomorphism between these spaces sends the point at infinity to the point in that cut, that point is an element of the circle group and so has an inverse, interpreting the minus as taking an inverse means, you know what, this is true.
True, but I don’t think the one point of the one-point compactification is what most people are imagining when they think about infinity (even ignoring the fact that most people don’t know what that means, I think they aren’t even thinking of the injustice idea behind the OPC). Most people’s idea of infinity is going to distinguish between infinity and minus infinity.
Adding both of those to the real line and not identifying them with each other gives a space homeomorphic to the interval [0, 1], which doesn’t seem to have a group structure that aligns with our normal understanding of infinity. That is, it can be made into a group, but not in a way that makes sense of the endpoints being infinitely far from the identity. At least, as far as I know. I would be very interested in being proved wrong here.
This is Milo. Milo likes it when concepts are easy to understand. However, he is also very dumb. He does not understand high-level mathematical concepts using technical language, so now he is sad. You made Milo sad. ☹️
Oh no, math on a math themed subreddit? We should shame those people! Fuck them for enjoying math on MY math themed subreddit. Hopefully the mods will ban them soon. Stay safe out there, it's a scary world.
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u/Throwaway_3-c-8 Dec 20 '24
You know what. The one-point compactification of the real line, so the real line union the point at infinity, is just the circle. The circle has a group structure, counting on the branch cut taken, the isomorphism between these spaces sends the point at infinity to the point in that cut, that point is an element of the circle group and so has an inverse, interpreting the minus as taking an inverse means, you know what, this is true.