I'm reading a book about algebraic topology recently and I have read through this sentence. "The space of of all one dimensional subspace is equal to the one dimensional circle (that's the circumference)" I don't understand this but there isn't a lot further explanation about this. Can anyone explain to me why it is like this? THANK YOU!~
1-dim subspace & sphere
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algebraic-topology
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0Yes. Space of all 1 dimensional subspaces of ℝ². – 2012-07-11
1 Answers
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The one dimensional subspaces are just the straight lines through the origin in $\mathbb R^n$
Each line through the origin is identified by a pair of antipodal points on the sphere $S^{n-1}$, the points being the points of intersection of the line with the sphere.
This correspondence is a one-one correspondence.
In $\mathbb R^2$, they are just the pairs of antipodal points in $S^1$.
Now the one dimensional spaces have been identified with antipodal pairs of points in $S^1$. The latter is quite easily seen to be homeomorphic to $S^1$ itself.
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1When $S^1$ is parametrized by the polar angle $\phi$ then the map $\phi\mapsto2\phi$ is a homeomorphism from $S^1/(\pm1)$ to $S^1$. – 2012-07-11