Degree of Discrete Generation of Compact Sets

In the present paper, under the continuum hypothesis, we construct an example of a discretely generated compact set $X$ whose square is not discretely generated. For each compact set $X$, there is an ordinally valued characteristic $\operatorname{idc}(X)$, which is the least number of iterations of the $d$-closure generating, as a result, the closure of any original subset $X$. We prove that if $\chi(X)\le\omega_\alpha$, then $\operatorname{idc}(X)\le\alpha+1$.