Mapping the Cantor ternary set onto compact metric spaces



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The heart of this paper is that any compact metric space is a continuous image of the Cantor ternary set. The heart of this function is a sequence of closed covers whose meshes converge to zero. Under the proper conditions functions with the Cantor set for domain are constructed to show that all totally disconnected compact metric spaces are homeomorphic. Connectedness of the range space does not preclude the existence of such a function, but it does imply that the function is not injective. An indecomposable continuum is also a continuous image of the Cantor set. The existence of an indecomposable continuum is shown. A bound on point inverse cardinalities, under this function, of limit points of a space whose dimension is less than or equal to one is established. When the point inverse cardinalities of this function are bounded then so is the dimension of the range space.



Metric spaces, Mappings (Mathematics)