In this paper we consider the space of all the linear semi-infinite programming problems with the same index set, endowed with a suitable topology. We provide a constructive proof of the following generic result: if we confine ourselves to the class of problems having a bounded set of coefficient vectors (those vectors appearing in the left-hand side of the constraints), the set of those problems which have a strongly unique optimal solution contains an open and dense subset of the set of solvable problems in the same class.