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 R.E. Hodel, "Arhangel'skii's Solution to Alexandroff's problem: A survey", Topol. Appl., 153(13) (2006) 2199-2217. 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.