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Completeness, Compactness, Effective Dimensions
Author(s) -
Binns Stephen
Publication year - 2013
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201100096
Subject(s) - mathematics , compact space , continuation , completeness (order theory) , metric space , metric (unit) , space (punctuation) , focus (optics) , convex metric space , discrete mathematics , pure mathematics , mathematical analysis , computer science , operations management , physics , optics , economics , programming language , operating system
We investigate a directed metric on the space of infinite binary sequences defined by\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ d(Y\rightarrow X)=\limsup _n\frac{C(X{\upharpoonright }n\Vert Y{\upharpoonright }n)}{n}, $$ \end{document} where C ( X ↾ n ‖ Y ↾ n ) is the Kolmogorov complexity of X ↾ n given Y ↾ n . In particular we focus on the topological aspects of the associated metric space—proving that it is complete though very far from being compact. This is a continuation of earlier work investigating other geometrical and toplogical aspects of this metric.