On Wallman compactifications of $T_0$-spaces and related questions

We study the compactification of the Wallman–Shanin type of $T_0$-spaces. We have introduced the notion of compressed compactification and proved that any compressed compactification is of the Wallman–Shanin type. The problem of the validity of the equality $\omega(X\times Y)=\omega X\times\omega Y$ is examined. Two open questions have arisen.