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

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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