On the weight-spectrum of a compact space (Q1261902)

For a space \(X\) \(Sp (w,X)\) denotes the set of weights of all infinite closed subspaces of \(X\); this set is called the weight spectrum of \(X\). The author studies the weight spectrum of compact Hausdorff spaces. If \(k\) is an uncountable regular cardinal and \(X\) is a compact \(T_ 2\) space with \(w(X) \geq k\), then \(Sp (w,X) \cap [k,2^{<k}] \neq \emptyset\). Two (known) corollaries of this theorem are: (1) every \(\omega\)- monolithic compact space with a small diagonal is metrizable; (2) under CH every compact space with a small diagonal is metrizable. The following result is both consistent with and independent of ZFC: For an infinite cardinal \(k\) with \(\omega_ 1 < k < c\) there is a separable compact LOTS \(X\) such that \(Sp(w,X) = \{\omega, k\}\). It is consistent with the continuum hypothesis that for every countable closed set \(T\) of cardinals \(<c\) there exists a separable compact LOTS \(X\) such that \(Sp (w,X) = \{\omega\} \cup T\).