I want to prove Jordan Curve Theorem doesnt hold for Torus ($T$) and Real Projective plane ($P$) ie I want to find embedding $f:S^1 \rightarrow S^1\times S^1 $ and $h:S^1 \rightarrow P$ such that $S^1\times S^1-f(S^1), P-h(S^1)$ are connencted.
Intuitivly I can see that. But I'm having trouble with rigourus proof. For instance this is what I have tried with the torus case:
I defined $f:S^1 \rightarrow S^1\times S^1 $ as $f(z) = (1,z)$ which is obiviously an embedding. But Im not sure how to show $ S^1\times S^1 - f(S^1) $ is connected. All I know is that I should get a space homeomorphic to cylinder.
Any advice?
In the case of $S^1 \times S^1$ and using $f$ with the formula given, you are right, the space $S^1 \times S^1 - f(S^1)$ is indeed homeomorphic to a cylinder. So your task, of course, is to prove this is so, by constructing a homeomorphism $$h : S^1 \times (0,1) \to (S^1 \times S^1) - f(S^1) $$ Perhaps if you write out the formulas for the domain and range, the formula for the required homeomorphism $h$ will be obvious. So, the domain is $$S^1 \times (0,1) = \{(z,t) \mid z \in S^1, t \in (0,1)\} $$ and $$(S^1 \times S^1) - f(S^1) = \{(z,w) \mid z \in S^1, w \in S^1 - \{1\}\} $$ Can you write down the formula now?