Research on computable structures of continuous spaces like the real number through representation in bottomed sequences

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The computational structure of the reals and other topological spaces are studied mainly with the method of embedding a space into the set of infinite sequences with bottoms. Some computationally natural properties of an embedding, such as the recursiveness of the definition and the non-redundancy of each digit in a code sequence, are investigated through their characterization in general topological terms, and some characterizations of spaces with such an embedding are given. Other topics such as constant time computability preserving conversions, and the relation between the Lawson topology of the space of formal balls and the hyperbolic topology of a metric space, are also investigated.