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Partial metric spaces. (English) Zbl 1229.54037
Summary: When mathematics is processed on a computer, objects are known only to the extent to which their values are computed; the metric space axiom that says \(d(x,x)=0\) for each point \(x\) then becomes the unrealistic assumption that we always know the eventual value of \(x\) exactly. The theory of partial metric spaces generalizes that of metric spaces by dropping that axiom to allow structures that simultaneously model mathematics and its computer representation. In them, \(d(x,x)=0\) for the ideal, completely known points; \(d(x,x)\neq 0\) for their partially computed approximations. We discuss how familiar metric and topological reasoning is refined to work in the general setting of convergence and continuity which can now be represented on computers.

MSC:
54E35 Metric spaces, metrizability
97I99 Analysis education
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