(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let X be the set of all points (x,y)[itex]\in[/itex]ℝ^{2}such that y=±1, and let M be the quotient of X by the equivalence relation generated by (x,-1)~(x,1) for all x≠0. Show that M is locally Euclidean and second-countable, but not Hausdorff.

2. Relevant equations

3. The attempt at a solution

So I haven't taken any topology and my understanding of it is to the basic. I've been reading a lot about quotient spaces and I'm really not understanding the phrase "let M be the quotient of X by the equivalence relation generated by (x,-1)~(x,1)"

Question: What is M? M={[x][itex]\in[/itex]X}={{v[itex]\in[/itex]X:v~x}:x[itex]\in[/itex]X}

Question: M is locally Euclidean because there is a one-to-one correspondence onto ℝ?

Question: M is second countable because the bases are open subsets of ℝ?

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# Topology, line with two origins

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