In topology, the split interval is a space that results from splitting each point in a closed interval into two adjacent points. It may be defined as the lexicographic product [0, 1] × {0, 1} with the order topology. 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.[1]
References
^Ostaszewski, A. J. (February 1974), "A Characterization of Compact, Separable, Ordered Spaces", Journal of the London Mathematical Societys2-7 (4): 758–760, doi:10.1112/jlms/s2-7.4.758
Further reading
Todorcevic, Stevo (6 July 1999), "Compact subsets of the first Baire class", Journal of the London Mathematical Society12 (4): 1179–1212, doi:10.1090/S0894-0347-99-00312-4
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