On a computable Hahn-Mazurkiewicz Theorem

The Aleksandrov-Hausdorff theorem states that a topological space is the continuous image of the Cantor set if and only if it is compact and metrizable. A computable version of this theorem is provided using the Type-2 Theory of Effectivity (TTE). In less formal terms, it is possible to write a computer program that can compute a surjection from the Cantor set C onto a computably compact computable metric space X with arbitrary precision. We also discuss work toward a computable version of the related Hahn-Mazurkiewicz theorem in this setting. Suppose it is possible to compute a Peano continuum, X, in the sense that it is possible to write a computer program that can draw (at least for the case where X is Euclidean) X with arbitrary precision. We wish to study the possibility of computing a surjection from [0, 1] onto X in the sense that it is possible to write a computer program that can approximate the values of this map with arbitrary precision.