Separation and cardinality - Some new results and old questions

It is well known that separation axioms together with some local and global cardinal invariants lead to restriction of the cardinality of a given topological space. For an extensive survey, one can look at [16]. We shall mention here just a few such results that are directly related to this paper. At the end we will discuss some long-standing open problems in cardinal invariant theory and give a possible way of approaching a solution.