Line with two origins

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This article is about a particular topological space (uniquely determined up to homeomorphism)|View a complete list of particular topological spaces
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


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.

Topological space properties

Properties it does not satisfy

Property Satisfied? Explanation Corollary properties satisfied/dissatisfied
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
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
locally compact space Yes
compact space No
second-countable space Yes satisfies: first-countable space, separable space
locally Euclidean space Yes satisfies: locally contractible space, locally Hausdorff space, locally normal space, locally regular space


Textbook references

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