One-dimensional locally connected S-spaces (Q1004043)

The authors construct, using Jensen's principle \(\diamondsuit\), a one-dimensional locally connected hereditarily separable continuum without convergent sequences. This improves a construction of the reviewer who constructed an infinite-dimensional locally connected continuum without convergent sequence from the Continuum Hypothesis. The example is the inverse limit of a carefully constructed inverse sequence of Menger universal curves. The authors also construct such spaces with noncoinciding dimensions. In fact, they show that there is such a space \(Z\) such that \(1 = \dim(Z) < {\text{\mathrm{ind}}}(Z) =\infty\). In addition, \(Z\) has the property that all of its perfect subsets are \(G_\delta\) sets, and \(Z\) is a strong \(S\)-space.