/sci/ - Science & Math

>>11530957
>that counterexample
My bad.
Consider [math]\mathbb{R} / \mathbb{Z}[/math] as the circle group [math]S^1[/math] and choose some irrational number [math]a \in [0, 1][/math]. Let [math]a[/math] generate a subgroup. It's a dense subset of [math]S^1[/math], naturally, and since it's not the entirety of [math]S^1[/math] (since it's countable, for example), it's not closed.
Now, the circle group is compact and Hausdorff. Thus, we can use this https://math.stackexchange.com/questions/83355/how-to-prove-that-a-compact-set-in-a-hausdorff-topological-space-is-closed to conclude it's not compact.

>>11497860
I don't think you can, because [math]f[/math] and [math]g[/math] are only specified to be continuous at [math]x_0[/math] .
I'm not sure if there was some version of that equivalence for functions only continuous at one point that I don't remember, tho.