/sci/ - Science & Math

>>16085973
I'm replying to myself but I think it is the case. Let [math]H(U)[/math] denote the collection of holomorphic functions on our region [math]U[/math]. Not really worth proving fully but you can embed [math]\mathbb{C}[/math] as a closed subset of [math]H(U)[/math] w.r.t. the topology I specified, and you can imagine how the one-point compactification of [math]H(U)[/math] restricted to this embedding would look a lot like [math]\hat{\mathbb{C}}[/math]. Pointwise projections from the compactification should force our one-point to take the value of infinity on all of [math]U[/math]... or something...
For those of you not in the know, when I say the topology of compact convergence I mean the one where a sequence of functions converges to whatever when it converges uniformly (w.r.t. the metric of whatever space these functions are mapping into) on all compact subsets of the region you're working on. Also generated by sets of the form [math]B(f,K,\varepsilon ):=\{ g:X\to Y \, | \, \sup_{x\in K} d_Y (f(x),g(x))<\varepsilon \} [/math] over all functions [math]f:X\to Y[/math], compact [math]K[/math], and all [math]\varepsilon >0[/math].
...But whatever, this is turbo-autism. Riddle me this /sqt/: should I bother attending my diffy geo class in the morning?