General Topology – Prove the Torus is Compact

compactnessgeneral-topology

I am having difficuties in showing a torus is compact. Initially I wanted to use Heine-Borel theorem, but after that I realise we are not working in $\mathbb{R}^n$ space. So a simple way to show torus is compact is by definition. But after defining an open cover for torus, I don't know how to proceed. Can anyone guide me?

Best Answer

If you are thinking of the torus as $S^1 \times S^1$:

  1. Products of compact sets are compact (you only need the finite case)
  2. $S^1$ compact (as it is a closed and bounded subset of $R^2$).

If you are thinking of the torus as $R^2 / Z^2$:

  1. This quotient map makes the same identifications as the exponential map $(e^{2 \pi i x}, e^{2 \pi i y})$, so since both are quotients $R^2 / Z^2$ is homeomorphic to $S^1 \times S^1$. (You will have to prove that if $q_1: X \to Y_1$ and $q_2 : X \to Y_2$ are quotient maps, and $q_1(x) = q_1(x')$ iff $q_2(x) = q_2(x')$, then $Y_1$ and $Y_2$ are (naturally) homeomorphic.)
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