Knowledge space theory and union-closed sets conjecture

The knowledge space theory provides a framework for knowledge management. One of major problems is to find core information for a body of knowledge. Union closed set conjecture, if true, guarantees that for a given knowledge space, there is an information that is linked to at least half of the knowledge units. This paper deals with a variant problem, where the knowledge space is also a topological space and possibly infinite. We prove that there is a point belonging to as many open sets as of the topological space itself.