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I have this problems

  1. Proof that the ball $B_1{(0,0)}$ can be embedded in Moore plane (Niemytzki plane)
  2. Proof that $({\mathbb R}^3; \textrm{usual topology})$ can be embedded in Moore plane (Niemytzki plane)
  3. Proof that $({\mathbb R}; \textrm{usual topology})$ can be embedded in Moore plane (Niemytzki plane)
    For the second I dont know if the function $h(x)=\left(x,\arctan(x)+ \frac{\pi}{2}\right)$ is right.

Thanks

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1 Answer 1

The first part should be easy via $(x,y)\mapsto (x,y+1)$, the third via $x\mapsto (x,1)$.

However, I doubt that the second statement is true: Assume $f\colon \mathbb R^3\to \Gamma$ is an embedding. If there is a point $(x,y)\in\mathbb R^3$ such that $f(p)=(u,v)$ has $v>0$, then there is an open 3D ball around $(x,y)$ that is embedded into an open disc in $\mathbb R^2$. This is not possible (though I'd need homology to show that). Therefore $f(\mathbb R^3)\subseteq \{(u,v)\in\Gamma\mid v=0\}$, but the latter set is discrete.

Edit: after the second part is changed from $\mathbb R^3$ to $\mathbb R^2$, the statement is clear: $\mathbb R$ is homeomorphic to an open interval, e.g. $(-1,1)$, hence $\mathbb R^2$ to $(0,1)^2$, which can be placed inside $\Gamma$.

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My fault it is ${\mathbb R}^2$ –  vic Oct 26 '12 at 17:11
    
Then please edit the question. –  Hagen von Eitzen Oct 26 '12 at 18:25
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