Here is the question I am trying to understand its solution:
Show that there exist nonorientable $1$-dimensional manifold if the Hausdorff condition is dropped from the definition of a manifold.
I know this question is here Non-orientable one dimensional manifold. but I am asking about a different thing:
What is the meaning of a line with 2 origins? Why it is not a hausdorff space?
I found an explanation online saying that the line with two origins is defined as follows: Take two copies of the real line, say $\mathbb R_1$ and $\mathbb R_2$ and identify every point in $\mathbb R_1$ with the corresponding point in $\mathbb R_2$ except for the origin.
Does this means the drawing of this space is as follows:
?
Take neighborhoods $N_1,N_2$ about $0_1,0_2$ respectively. They must intersect as they both contain some small $\epsilon>0$. Thus the space is not Hausdorff as the copies of zero cannot be separated by open sets.