In topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

, the split interval is a space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

 that results from splitting each point in a closed interval into two adjacent points. It may be defined as the lexicographic product

Lexicographical order

In mathematics, the lexicographic or lexicographical order, , is a generalization of the way the alphabetical order of words is based on the alphabetical order of letters.-Definition:Given two partially ordered sets A and B, the lexicographical order on...

 [0, 1] × {0, 1} with the order topology

Order topology

In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets...

. It is also known as the Alexandrov double arrow space or two arrows space.

The split interval is compact Hausdorff, and it is hereditarily Lindelöf and hereditarily separable, but it is not metrizable; its metrizable subspaces are all countable.

All compact, separable ordered spaces are order-isomorphic to a subset of the split interval.