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.

The split interval can be defined as the lexicographic product ${\displaystyle [0,1]\times \{0,1\}}$ equipped with the order topology.^[1] Equivalently, the space can be constructed by taking the closed interval ${\displaystyle [0,1]}$ with its usual order, splitting each point ${\displaystyle a}$ into two adjacent points ${\displaystyle a^{-}<a^{+}}$, 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, ${\displaystyle (0,0)}$ and ${\displaystyle (1,1)}$ 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 ${\displaystyle 0}$ and ${\displaystyle 1}$ 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 ${\displaystyle ((a,b]\times \{0\})\cup ([a,b)\times \{1\})}$ with ${\displaystyle a<b}$. (In the point splitting description these are the clopen intervals of the form ${\displaystyle [a^{+},b^{-}]=(a^{-},b^{+})}$, which are simultaneously closed intervals and open intervals.) The lower subspace ${\displaystyle (0,1]\times \{0\}}$ is homeomorphic to the Sorgenfrey line with half-open intervals to the left as a base for the topology, and the upper subspace ${\displaystyle [0,1)\times \{1\}}$ 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.

The split interval ${\displaystyle X}$ 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 ${\displaystyle X\times X}$ 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]

• List of topologies – List of concrete topologies and topological spaces

• 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) [1978]. Counterexamples in Topology (Dover reprint of 1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 0507446.