Simultaneous Extensions of Topologies Through Traces of Neighbourhood Filters

Let X be a set, X_i X for i I and, for x X, a filter in X_i. The paper gives necessary and sufficient conditions for the existence of a topology on X compatible with , i.e. such that is the trace on X_i of the -neighbourhood filter of x. It is shown that, among these compatible topologies,there are always a coarsest one and a finest one. Some applications are also given.