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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the Szpilrajn extension theorem, due to Edward Szpilrajn (Edward Marczewski) in 1930, is one of many examples of the use of the axiom of choice (in the form of Zorn''s lemma) to find a maximal of a set with certain given properties. The theorem states that, given a binary relation R that is irreflexive and transitive it is always possible to find an extension of the relation (i.e. a relation T that strictly includes R) which is asymmetric, negatively transitive and connected. First of all, we need some definitions to be clear upon the terminology we will use speaking about relations with particular properties.