Line with two origins

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This article describes a standard counterexample to some plausible but false implications. In other words, it lists a pathology that may be useful to keep in mind to avoid pitfalls in proofs
View other standard counterexamples in topology

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 $\mathbb{R}$, 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.

Topological space properties

Properties it does not satisfy

• Hausdorff space: The line with two origins is not a Hausdorff space, because we cannot find open sets separating the two origins
• US-space: it is not true for this space that every sequence has at most one limit

Properties it does satisfy

• Locally Euclidean space: Every point has a neighbourhood homeomorphic to ${\mathbb{R}}^1$
• Second-countable space: There is a countable basis of open sets

Thus the line with two origins fails to be a manifold only with respect to the Hausdorff condition. It behaves very differently from what we intuitively expect of manifolds