Rigid continua and transfinite inductive dimension

We introduce a general method of resolving first countable, compact spaces that allows accurate estimate of inductive dimensions. We apply this method to construct, inter alia, for each ordinal number α>1 of cardinality ?c, a rigid, first countable, non-metrizable continuum Sα with . Sα is the increment in some compactification of 0,1) and admits a fully closed, ring-like map onto a metric continuum. Moreover, every subcontinuum of Sα is separable. Additionally, Sα can be constructed so as to be: (1) a hereditarily indecomposable Anderson-Choquet continuum with covering dimension a given natural number n, provided α>n, (2) a hereditarily decomposable and chainable weak Cook continuum, (3) a hereditarily decomposable and chainable Cook continuum, provided α is countable, (4) a hereditarily indecomposable Cook continuum with covering dimension one, or (5) a Cook continuum with covering dimension two, provided α>2.We also produce a chainable and hereditarily decomposable space Sω(c⁺) with , , trind₀Sω(c⁺) and trInd₀Sω(c⁺) all equal to ω(c⁺), the first ordinal of cardinality c⁺.