Fun little math problem of the day
This little fact was mentioned without proof in one of my math classes a few weeks ago, but it just occurred to me today to post it here. The proof I came up with is, in my opinion, pretty pretty (but given the reaction of some people at mathcamp when I said I thought pointset topology was pretty, you might disagree).
Prove that the usual topology on R^n is the unique topology that makes it both a topological vector space and a locally compact Hausdorff space. Here, "topological vector space" means that addition is continuous and that scalar multiplication is continuous as a map from R \times R^n to R^n, where R is given the usual topology.
Prove that the usual topology on R^n is the unique topology that makes it both a topological vector space and a locally compact Hausdorff space. Here, "topological vector space" means that addition is continuous and that scalar multiplication is continuous as a map from R \times R^n to R^n, where R is given the usual topology.