on E correspond as
follows:
One calls an ``Alexandrov topology'' a topological space where an arbitrary intersection of open sets is open, or equivalently, every point has a least neighbourhood. Alexandrov called it a ``discrete space''.
The Alexandrov specialization theorem characterizes an Alexandrov
topology in terms of a quasi-order, that is, a reflexive and
transitive relation on that space. An Alexandrov topology on E and a
quasiorder on E correspond as
follows:
q iff p belongs to the
closure of {q};
, that is, sets X with (p 
X, p q) implies
(q X).
Moreover that quasiorder is an order relation (i.e., it is antisymmetric) iff the topology satisfies the T0 separation axiom.
The usual reference for this result is [2], as well as Alexandrov's later works (such as his topology book with Hopf, or his monograph on combinatorial topology).
In fact, while developing his theory, Alexandrov discussed with Birkhoff. In [1] Alexandrov proposed the axioms for a ``discrete space'' and stated that it corresponded to a partially ordered set. Birkhoff answered him in [3], and among other things stated that he was studying similar structures in the framework of abstract algebra. Then Alexandrov published his results in [2], in particular the specialization theorem. The same year Birkhoff published [4], where he proved that a ``complete ring of sets'' (family of sets closed under arbitrary union and intersection, including void ones) corresponds to a quasiorder. A ``complete ring of sets'' means exactly the same thing as ``Alexandrov topology'', and the quasiorder given by Birkhoff is the same as the one given by Alexandrov, but in another wording.
Birkhoff showed also that the ``complete ring of sets'' is a ``complete field of sets'' (the family is also closed under complementation) iff the quasiorder is an equivalence relation (i.e., it is symmetric); in topological terms, it means that the open sets coincide with the closed sets.
Therefore the specialization theorem attributed to Alexandrov is in some way also due to Birkhoff.
[1] Paul Alexandrov, Sur les espaces discrets, Comptes Rendus de
l'Académie des Sciences 200 (1935), 1469-1471.
[2] Paul Alexandrov, Diskrete Räume, Mat. Sb. 2 (1937), no. 44,
501-519.
[3] Garrett Birkhoff, Sur les espaces discrets, Comptes Rendus de
l'Académie des Sciences 201 (1935), 19-20.
[4] Garrett Birkhoff, Rings of sets, Duke Math J. 3 (1937), no. 3,
443-454.
March the 29th, 2007
Christian Ronse