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
[1] Equivalently, the space can be constructed by taking the closed interval
with its usual order, splitting each point
Some authors[5][6] take as definition the same space without the two isolated points.
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
(In the point splitting description these are the clopen intervals of the form
is homeomorphic to the Sorgenfrey line with half-open intervals to the left as a base for the topology, and the upper subspace
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.
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.
All compact, separable ordered spaces are order-isomorphic to a subset of the split interval.