Split interval
In topology, the split interval, or double arrow space, is a topological space that results from splitting each point in a closed interval into two adjacent points and giving the resulting ordered set the order topology. It satisfies various interesting properties and serves as a useful counterexample in general topology.
Definition
The split interval can be defined as the lexicographic product [math]\displaystyle{ [0, 1] \times\{0, 1\} }[/math] equipped with the order topology.[1] Equivalently, the space can be constructed by taking the closed interval [math]\displaystyle{ [0,1] }[/math] with its usual order, splitting each point [math]\displaystyle{ a }[/math] into two adjacent points [math]\displaystyle{ a^-\lt a^+ }[/math], and giving the resulting linearly ordered set the order topology.[2] The space is also known as the double arrow space,[3][4] Alexandrov double arrow space or two arrows space.
The space above is a linearly ordered topological space with two isolated points, [math]\displaystyle{ (0,0) }[/math] and [math]\displaystyle{ (1,1) }[/math] in the lexicographic product. Some authors[5][6] take as definition the same space without the two isolated points. (In the point splitting description this corresponds to not splitting the endpoints [math]\displaystyle{ 0 }[/math] and [math]\displaystyle{ 1 }[/math] of the interval.) The resulting space has essentially the same properties.
The double arrow space is a subspace of the lexicographically ordered unit square. If we ignore the isolated points, a base for the double arrow space topology consists of all sets of the form [math]\displaystyle{ ((a,b]\times\{0\}) \cup ([a,b)\times\{1\}) }[/math] with [math]\displaystyle{ a\lt b }[/math]. (In the point splitting description these are the clopen intervals of the form [math]\displaystyle{ [a^+,b^-]=(a^-,b^+) }[/math], which are simultaneously closed intervals and open intervals.) The lower subspace [math]\displaystyle{ (0,1]\times\{0\} }[/math] is homeomorphic to the Sorgenfrey line with half-open intervals to the left as a base for the topology, and the upper subspace [math]\displaystyle{ [0,1)\times\{1\} }[/math] is homeomorphic to the Sorgenfrey line with half-open intervals to the right as a base, like two parallel arrows going in opposite directions, hence the name.
Properties
The split interval [math]\displaystyle{ X }[/math] is a zero-dimensional compact Hausdorff space. It is a linearly ordered topological space that is separable but not second countable, hence not metrizable; its metrizable subspaces are all countable.
It is hereditarily Lindelöf, hereditarily separable, and perfectly normal (T6). But the product [math]\displaystyle{ X\times X }[/math] of the space with itself is not even hereditarily normal (T5), as it contains a copy of the Sorgenfrey plane, which is not normal.
All compact, separable ordered spaces are order-isomorphic to a subset of the split interval.[7]
See also
- List of topologies – List of concrete topologies and topological spaces
Notes
- ↑ Todorcevic, Stevo (6 July 1999), "Compact subsets of the first Baire class", Journal of the American Mathematical Society 12: 1179–1212, doi:10.1090/S0894-0347-99-00312-4
- ↑ Fremlin, section 419L
- ↑ Arhangel'skii, p. 39
- ↑ Ma, Dan. "The Lexicographic Order and The Double Arrow Space". https://dantopology.wordpress.com/2009/10/07/the-lexicographic-order-and-the-double-arrow-space.
- ↑ Steen & Seebach, counterexample #95, under the name of weak parallel line topology
- ↑ Engelking, example 3.10.C
- ↑ Ostaszewski, A. J. (February 1974), "A Characterization of Compact, Separable, Ordered Spaces", Journal of the London Mathematical Society s2-7 (4): 758–760, doi:10.1112/jlms/s2-7.4.758
References
- Arhangel'skii, A.V. and Sklyarenko, E.G.., General Topology II, Springer-Verlag, New York (1996) ISBN 978-3-642-77032-6
- Engelking, Ryszard, General Topology, Heldermann Verlag Berlin, 1989. ISBN:3-88538-006-4
- Fremlin, D.H. (2003), Measure Theory, Volume 4, Torres Fremlin, ISBN 0-9538129-4-4
- Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995). Counterexamples in Topology (Dover reprint of 1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3.
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