 The Galois connections described in the opening section are what are sometimes called contravariant (as opposed to covariant) Galois connections; they were introduced in somewhat different terminology by Garrett Birkhoff (1967). Closure operations as defined in the opening section are sometimes called closure operators. Note that the definition given, although standard, is much less restrictive than the notion of a specifically topological closure operation (in particular, not requiring C(X ∪ Y) = C(X) ∪ C(Y) or C(∅) = ∅,). Some authors discuss matters of logical theory placing a particular emphasis on closure operations and their systems of closed sets (sets X with C(X) = X); see Pollard and Martin (1996) and references there cited.