- #1

BrainHurts

- 102

- 0

## Homework Statement

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.

## Homework Equations

## 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?