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Quasi-metric properties of complexity spaces. (English) Zbl 0941.54028
Recall that a quasi-uniform space $(X,{\cal U})$ is bicomplete iff the uniform space $(X,{\cal U}^s)$ is complete, where ${\cal U}^s:={\cal U}\vee{\cal U}^{-1}$; it is Smyth-completeable iff every left K-Cauchy filter on $(X,{\cal U})$ is a Cauchy filter on $(X,{\cal U}^s)$. The complexity space $C:= \{f: \omega\to (0,+\infty]$; $\sum (2^{-n} f(n)^{-1}< \infty\}$ with the quasi metric $\rho(f,g):= \sum 2^{-n} \max\{ g(n)^{-1}- f(n)^{-1},0\}$ and its dual space $C^*:= \{f:\to \bbfR$; $\sum 2^{-n} f(n)<+\infty\}$ with the quasi-metric $\rho^*(f,g):= \sum 2^{-n} \max\{g(n)- f(n),0\}$ are discussed in this paper. It is shown that the quasi-uniform space associated with $(C^*, \rho^*)$ is Smyth-complete and consequently is a Baire space. For a closed ${\cal F}\subseteq C^*$ and for $m\in C^*$ the space ${\cal F}_m:= \{f\in{\cal F}$; $m$ is an upper bound for $f\}$ with metric $\rho^s(f,g):= \rho^*(f,g)+ \rho^*(g,f)$ turns out to be a compact metric space. The paper is nicely written, but the reader (like the reviewer) would have appreciated a little more self-containedness. The computational significance of the results are far from being clear and might have been pointed out in greater detail by the authors.

MSC:
54E15Uniform structures and generalizations
54C30Real-valued functions on topological spaces
54E35Metric spaces, metrizability
54C35Function spaces (general topology)
68Q25Analysis of algorithms and problem complexity
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