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 $x\neq 0$.
• 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

Property	Satisfied?	Explanation	Corollary properties satisfied/dissatisfied
Separation
locally Hausdorff space	Yes		satisfies: T1 space, Kolmogorov space
US-space	No		dissatisfies: KC-space, Hausdorff space, regular space (note, it's T1 so regular would imply Hausdorff), normal space, metrizable space, CW-space
locally normal space	Yes
locally regular space	Yes
Connectedness
path-connected space	Yes		satisfies: connected space
locally contractible space	Yes		satisfies: locally path-connected space, locally simply connected space, semilocally simply connected space, locally connected space
Compactness
locally compact space	Yes
Countability
second-countable space	Yes		satisfies: first-countable space, separable space
Miscellaneous
locally Euclidean space	Yes		satisfies: locally contractible space, locally Hausdorff space, locally normal space, locally regular space

References

Textbook references

• Topology (2nd edition) by James R. Munkres^{More info}, Page 227, Exercise 5, Chapter 4 (full definition given in exercise, by specifying a basis)