Math 2.0

General topology Exotic spaces and counterexamples Counterexample

Line with two origins

Builds on Hausdorff spaceSecond-countable spaceConvergent sequenceNeighbourhood

Counterexample

Let Y=ℝ×{0,1}Y = \R \times \set{0, 1}, the disjoint union of two copies of the real line. Identify (x,0)(x, 0) with (x,1)(x, 1) for every x≠0x \ne 0, and give the set XX of classes the quotient topology. The two classes 0a0_a of (0,0)(0, 0) and 0b0_b of (0,1)(0, 1) are the two origins. The space XX is locally homeomorphic to ℝ\R, second-countable and T1T_1, but it is not Hausdorff.

Remarks

Removing 0b0_b leaves an open set homeomorphic to ℝ\R, and the same holds for 0a0_a; these two open sets cover XX. Every neighbourhood of 0a0_a contains the image of (−ε,0)∪(0,ε)(-\varepsilon, 0) \cup (0, \varepsilon) for some ε>0\varepsilon > 0, and so does every neighbourhood of 0b0_b. So any neighbourhood of 0a0_a meets any neighbourhood of 0b0_b, and the two origins cannot be separated. For the same reason the sequence of classes of 1/n1/n converges to both 0a0_a and 0b0_b, so limits of sequences are not unique. This is why the usual definition of a topological manifold requires the Hausdorff condition: without it, this space would count as a 11-manifold.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Convergent sequence, concept, no evidence levelConvergent sequenceconceptHausdorff space, concept, no evidence levelHausdorff spaceconceptNeighbourhood, concept, no evidence levelNeighbourhoodconceptSecond-countable space, concept, no evidence levelSecond-countable spaceconceptLine with two origins, counterexample, no evidence levelLine with two originscounterexample
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Convergent sequence concept
  2. Hausdorff space concept
  3. Neighbourhood concept
  4. Second-countable space concept
  5. Line with two origins counterexample

Further reading

  1. Web https://en.wikipedia.org/wiki/Non-Hausdorff_manifold, Line with two origins
About this page
TagE2K0MM2MQQ
URIm2://E2K0MM2MQQ
Kindcounterexample
WritingW0 stub
MSC54G20, 54D10, 57N99
UsesHausdorff space, Second-countable space, Convergent sequence, Neighbourhood
Changed2 November 2026

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