Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In topology, an Alexandrov space is a topological space in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any finite family of open sets is open. In an Alexandrov space the finite restriction is strengthened. Alexandrov topologies are uniquely determined by their specialization preorders. Indeed, given any preorder <= on a set X, there is a unique Alexandrov topology on X for which the specialization preorder is <=. The open sets are just the upper sets with respect to <=. Thus, Alexandrov topologies on X are in one-to-one correspondence with preorders on X.

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