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 S α with trind S α = trInd S α = trind 0 S α = trInd 0 S α = α . S α 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 S α is separable. Additionally, S α 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 . o produce a chainable and hereditarily decomposable space S ω ( c + ) with trind S ω ( c + ) , trInd S ω ( c + ) , trind 0 S ω ( c + ) and trInd 0 S ω ( c + ) all equal to ω ( c + ) , the first ordinal of cardinality c + .