Fully closed maps and non-metrizable higher-dimensional Anderson–Choquet continua

Abstract

Abstract. Fedorchuk's fully closed (continuous) maps and resolutions are applied in constructions of non-metrizable higher-dimensional analogues of Anderson, Choquet, and Cook's rigid continua. Certain theorems on dimensionlowering maps are proved for inductive dimensions and fully closed maps from spaces that need not be hereditarily normal, and some examples of continua have non-coinciding dimensions.Fully closed (continuous) maps and resolutions appear in numerous constructions (see S. Watson [48], V.V. Fedorchuk [26] for surveys), in particular, in constructions of homogeneous spaces with non-coinciding dimensions. In this paper we apply the maps in order to obtain examples of continua with strong, hereditary rigidity properties.A non-degenerate continuum X is called -an Anderson-Choquet continuum if every non-degenerate subcontinuum P of X has exactly one embedding P → X, the identity id P ; -a Cook continuum if every non-degenerate subcontinuum P of X has exactly one non-constant map P → X, the identity id P .Examples were constructed-respectively-by R. [32] for more references). In [32] the present author constructed a metric, n-dimensional (arbitrary n > 1), hereditarily indecomposable continuum no two of whose disjoint n-dimensional subcontinua are homeomorphic, but he was not able to ensure that the continuum be Anderson-Choquet.2000 Mathematics Subject Classification. 54F15, 54F45.

Components and inductive dimensions of compact spaces

“…On the other hand, the present author [13] has proved certain theorems on dimension-lowering maps for Ind, for Charalambous-Filippov-Ivanov inductive dimension Ind 0 (M. G. Charalambous [2], A. V. Ivanov [12]), and for fully closed maps from spaces that need not be hereditarily normal (see Section 3 in the present paper).…”

On Dimensionsgrad, resolutions, and chainable continua

“…If in the above definition of Ind, we stipulate that the set A is a singleton, we obtain the definition of ind. In [5,6,22] we used Ind 0 as a tool for estimating ind and Ind. Dg or Dimensionsgrad is Brouwer's original definition of dimension [3].…”