Line with two origins

Definition
The line with two origin can be defined in the following equivalent ways:


 * It is the quotient space of the disjoint union of two copies of $$\R$$, via the identification of $$x$$ in the first copy with $$x$$ in the second copy, for $$x \ne 0$$.
 * It is the real line with two origins, i.e. with two copies of the origin, wherein although both copies of the origin are separated, arbitrarily small nonzero real numbers approach both these copies.
 * It is the topological space with an étale map to the real line, such that the fiber at the origin is a two-point set, and the fiber at any other point of the real line is a one-point set.

Textbook references

 * , Page 227, Exercise 5, Chapter 4 (full definition given in exercise, by specifying a basis)