The authors consider here some peculiar properties of Tychonoff spaces \(X\) and corresponding real function spaces \(C(X)\), exhibiting connections between them. The problems discussed concern the so-called property (db): if \(E\) is the union of an increasing sequence \((E_n)\) of vector spaces, then some \(E_n\) is dense and barreled. The authors solve here the long standing problem of characterizing those Tychonoff spaces \(X\) for which the function space \(C_k(X)\) with the compact-open topology satisfies property (db). The authors consider also some results related to strongly Hewitt spaces, give three examples of spaces with special properties and some questions and problems.