
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.

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 $A$.

Re: 2 topology problems
Sorry, typo – should be intersect.