Recently I have found such a thing like ,,finite boolean combination of open sets" (of a topological space). Unfortunatlety I haven't found anywhere the precise definition of such combination.
Does anyone know where could I find it? The only thing that I can figure out myself is connected with the boolean algebra of sets where we have three operations: union, intersection and complement of sets. Is this combination a set that we can get using intersections (of two sets), unions (also of two sets) and complement operations applied finitely many times to the mentioned family of open sets? Am I right?
Your intuition is correct, and the notion refers to the concrete Boolean algebra of subsets of a given set, also (confusingly) called a field of sets.
The operations in a Boolean algebra (AND, OR, NOT) are often called combinations. Maybe this is because originally logic gates were combined to make more complicated circuits.
The set operations complement, union, intersection directly correspond to logical combinations of the membership property. That is, the statement $x\in X\cap Y$ translates to $x\in X$ AND $x\in Y$ and so on. Therefore, the term "Boolean combinations" applies to both combining logical formulas with AND, OR, NOT and sets with intersection, union, and complement.