Chains of spaces (Q689612)

A partial ordering \(\vartriangleleft\) of the class of topological spaces is considered: \(X\vartriangleleft Y\) if \(X\) is homeomorphic to a nowhere dense closed subset of \(Y\). It is shown that for every linear ordering \(A\) of cardinality at most continuum there exists a chain of copies of \(\beta \omega\) isomorphic to \(A\). This theorem generalizes the earlier result of the author [Colloq. Math. Soc. Janos Bolyai 41, 261-267 (1985; Zbl 0615.54003)] for \(A={\mathfrak c}\) with its natural order.