Split interval

In topology, the split interval is a topological space that results from splitting each interior point in a closed interval into two adjacent points. It is also known as the Alexandrov double arrow space or two arrows space. 
Definition.
It may be defined as the lexicographic product [0, 1] × {0, 1} without the isolated edge points, (0,1) and (1,0), equipped with the order topology. 
Properties.
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.