Generalizations of two cardinal inequalities of Hajnal and Juhász

In this paper we also generalize (2) by showing that if X is a T1-space, then |X|≤2nhs(X)⋅2s(X). It follows from our result that (2) is true for every T1-space for which nhs(X)≤2s(X). A simple example shows that the presence of the cardinal function nhs(X) in our inequality is essential.