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Math Help - 2 topology problems

  1. #1
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    2 topology problems

    Hi. Can anybody help me with this problems:

    1. Prove that S^1 \ N is homeomorphic with R. S^1 = {(x,y)|x^2 + y^2=1}, N=(0,1) (S^1\N is circular without one point, (0,1))

    2. If set A is countable subset of R^2 (of plane), prove that R^2 \ A is path connected.
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  2. #2
    Member GLaw's Avatar
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    Re: 2 topology problems

    1. Note that $S=\{(\sin\theta,\cos\theta):0 < \theta < 2\pi\}$ so it is homeomorphic to the open interval $(0,\,2\pi)$. Define $f:(0,\,2\pi)\to\mathbb R$ by $f(\theta)=\tan\left(\dfrac{\theta-\pi}2\right)$.

    2. Hint: Given any two points in $\mathbb R^2$, there are uncountably many continuous paths between them, and only a countable number of these paths instersect $A$.
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  3. #3
    Member GLaw's Avatar
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    Re: 2 topology problems

    Sorry, typo – should be intersect.
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