What makes a space have large weight?

We formulate several conditions (two of them are necessary and sufficient) which imply that a space of small character has large weight. In section 3 we construct a ZFC example of a first countable 0-dimensional space X of size 2^omega with w(X)=2^omega and nw(X)=omega, we show that CH implies the existence of a 0-dimensional space Y of size omega_1 with w(Y)=nw(Y)=omega_1 and chi(Y)=R(Y)=omega, and we prove that it is consistent that 2^omega is as large as you wish and there is a 0-dimensional space Z of size 2^omega such that w(Z)=nw(Z)=2^omega but chi(Z)=R(Z^omega)=omega

Two improvements on Tkačenko's addition theorem

summary:We prove that (A) if a countably compact space is the union of countably many $D$ subspaces then it is compact; (B) if a compact $T_2$ space is the union of fewer than $N(\Bbb R)$ = \operatorname{cov} (\Cal M) left-separated subspaces then it is scattered. Both (A) and (B) improve results of Tkačenko from 1979; (A) also answers a question that was raised by Arhangel'ski\v{i} and improves a result of Gruenhage