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Let $\beta\omega$ be the Stone-Čech compactification of the discrete infinite countable space $\omega$, and $\beta^*\omega=\beta\omega\smallsetminus \omega$ is the Stone-Čech remainder. The map $j:n\...

Since there seems to be no progress in this interesting question, I took the liberty to reformulate it in a way, that is easier to understand. Moreover, my answer shows that the question is related to ...