Line with two origins
From Topospaces
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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
Contents
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 , via the identification of in the first copy with in the second copy, for .
- 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 | ||
compact space | No | ||
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)