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I am trying to find a Hausdorff topological space that is not second-countable but otherwise a DIFFERENTIABLE n-manifold. I can't figure it out. Does it exist?
I read about the classical example of [tex]L=\omega_1\times[0,1)[/tex] with lexicographical order and the order topology. It's Hausdorff, not second-countable and locally homeomorphic to [tex]\mathbb{R}[/tex]. (found a nice http://www.uoregon.edu/~koch/math431/LongLine.pdf" ) To make an n-manifold I thought [tex]L\times[0,1]^{n-1}[/tex] could work. But is [tex]L[/tex] a differentiable manifold? Are the gluing maps [tex]C^\infty[/tex]? Can the maps be constructed so that they are?
If this doesn't work I have no clue what could it be. Does anyone know an example?
I read about the classical example of [tex]L=\omega_1\times[0,1)[/tex] with lexicographical order and the order topology. It's Hausdorff, not second-countable and locally homeomorphic to [tex]\mathbb{R}[/tex]. (found a nice http://www.uoregon.edu/~koch/math431/LongLine.pdf" ) To make an n-manifold I thought [tex]L\times[0,1]^{n-1}[/tex] could work. But is [tex]L[/tex] a differentiable manifold? Are the gluing maps [tex]C^\infty[/tex]? Can the maps be constructed so that they are?
If this doesn't work I have no clue what could it be. Does anyone know an example?
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