Embeddings into pseudocompact spaces of countable tightness. (Q1426485)

Under the existence of scales of cardinality \(\omega_1\) which is implied by CH, the following important result is proved. Let \({\mathfrak F}\) be a free filter on \(\omega\). We assume that \({\mathfrak F}\) has a base which is well ordered by \(\subset^*\) of type \(\omega_1\). Then there exists a pseudocompact \(0\)-dimensional \(T_1\)-space of countable tightness which has a closed subspace homeomorphic to \(\omega\cup\{{\mathfrak F}\}\). The symbol \(\subset^*\) means \(B- A\) is finite. Moreover, two existence theorems are stated under CH that are concerned with pseudocompact Tikhonov spaces of countable tightness. If \(p\in \beta\omega- \omega\) is a P-point (i.e., if every \(G_\delta\) which contains a point \(a\) is a neighborhood of \(a\), then \(a\) is a P-point), then there exists a pseudocompact Tikhonov space of countable tightness which contains a closed copy of \(\omega\cup\{p\}\subset \beta\omega\). \(\beta\omega\) is the Čech compactification of \(\omega\). Another one is: there exists a pseudocompact Tikhonov space of countable tightness which contains a non-discrete extremally disconnected subspace.