# general topology – \$S^{1}\$ is locally euclidean

Prove that the subspace $$S^{1}={{(x,y)in mathbb{R^{2}} : d((x, y),(0,0))=1}}$$ with $$d$$ the euclidien metric on $$mathbb{R^{2}}$$ is locally euclidean..
Let $$(x,y)in S^{1}$$ such that $$(x,y)not= (1,0)$$.
Then $$(x,y)in S^{1}setminus{{(1,0)}}=mathbb{R^{2}}setminus{(1,0)}bigcap S^{1}$$ open in S^{1} and we now that $$S^{1}setminus{{(1,0)}}$$ is homeomorphic to $$)0,2pi($$.
Now for {(1,0)} I can’t seem to find an open subset of S^{1} that contains (1,0) and homeomorphic to some open subset of $$mathbb{R}$$..any help please.