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.
== Definition ==
The '''split interval''' can be defined as the lexicographic product <math>[0, 1] \times\{0, 1\}</math> equipped with the order topology.<ref>{{Citation |last=Todorcevic |first=Stevo |authorlink=Stevo Todorčević|date=6 July 1999 |title=Compact subsets of the first Baire class |journal=Journal of the American Mathematical Society |volume=12 |issue=4 |pages=1179–1212 |doi=10.1090/S0894-0347-99-00312-4|doi-access=free }}</ref> Equivalently, the space can be constructed by taking the closed interval <math>[0,1]</math> with its usual order, splitting each point <math>a</math> into two adjacent points <math>a^-<a^+</math>, and giving the resulting linearly ordered set the order topology.<ref>Fremlin, section 419L</ref> The space is also known as the '''double arrow space''',<ref>Arhangel'skii, p. 39</ref><ref>{{cite web |last1=Ma |first1=Dan |title=The Lexicographic Order and The Double Arrow Space |date=8 October 2009 |url=https://dantopology.wordpress.com/2009/10/07/the-lexicographic-order-and-the-double-arrow-space}}</ref> '''Alexandrov double arrow space''' or '''two arrows space'''.
The space above is a linearly ordered topological space with two isolated points, <math>(0,0)</math> and <math>(1,1)</math> in the lexicographic product. Some authors<ref>Steen & Seebach, counterexample #95, under the name of '''weak parallel line topology'''</ref><ref>Engelking, example 3.10.C</ref> take as definition the same space without the two isolated points. (In the point splitting description this corresponds to not splitting the endpoints <math>0</math> and <math>1</math> 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 <math>((a,b]\times\{0\}) \cup ([a,b)\times\{1\})</math> with <math>a<b</math>. (In the point splitting description these are the clopen intervals of the form <math>[a^+,b^-]=(a^-,b^+)</math>, which are simultaneously closed intervals and open intervals.) The lower subspace <math>(0,1]\times\{0\}</math> is homeomorphic to the Sorgenfrey line with half-open intervals to the left as a base for the topology, and the upper subspace <math>[0,1)\times\{1\}</math> 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.
== Properties ==
The split interval <math>X</math> 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 (T<sub>6</sub>). But the product <math>X\times X</math> of the space with itself is not even hereditarily normal (T<sub>5</sub>), 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.<ref>{{Citation |last=Ostaszewski |first=A. J. |date=February 1974 |title=A Characterization of Compact, Separable, Ordered Spaces |journal=Journal of the London Mathematical Society |volume=s2-7 |issue=4 |pages=758–760 |doi=10.1112/jlms/s2-7.4.758}}</ref>
== See also ==
* {{annotated link|List of topologies}}
== Notes == {{reflist}}
== References ==
* 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}} * {{citation|first=D.H.|last=Fremlin|title=Measure Theory, Volume 4|publisher=Torres Fremlin|year=2003|isbn=0-9538129-4-4}} * {{Cite book | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=Counterexamples in Topology | orig-year=1978 | publisher=Springer-Verlag | location=Berlin, New York | edition=Dover reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 | year=1995 }}
Category:Topological spaces