Quasi-metric properties of complexity spaces

The complexity (quasi-metric) space has been introduced as a part of the development of a topological foundation for the complexity analysis of algorithms (Schellekens, 1995). Applications of this theory to the complexity analysis of Divide and Conquer algorithms have been discussed by Schellekens (1995). e obtain several quasi-metric properties of the complexity space. The main results obtained are the Smyth-completeness of the complexity space and the compactness of closed complexity spaces which possess a (complexity) lower bound. Finally, some implications of these results in connection to the above mentioned complexity analysis techniques are discussed and the total boundedness of complexity spaces with a lower bound is discussed in the light of Smythʹs computational interpretation of this property (Smyth, 1991).