ONT Re: Prospects for Inquiry Driven Systems

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PRO.  Note 49

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2.2.1.2.  Topology and Metric

Topology is the most unconstrained study of spaces, beginning as it does
with spaces that have barely enough hope of geometric structure to deserve
the name of spaces (Kelley, 1961).  An attention to this discipline inspires
caution against taking too lightly the issue of a metric.  There is no longer
any reason to consider the question of a metric to be a trivial one, something
whose presence and character can be taken for granted.  For each space that can
be contemplated there arises a typical suite of questions about the existence
and the uniqueness of a possible metric.  Some spaces are not metrizable at
all (Munkres, sec. 2-9).  Those that are may have a multitude of different
metrics defined on them.  My own sampling of differential methods in AI,
both smooth and chunky style, suggests to me that this multiplicity of
possible metrics is the ingredient that conditions one of their chief
sticking points, a computational viscosity that consistently sticks
in the craw of computers.  Unpalatable if not intractable, it will
continue to gum up the works, at least until some way is found to
dissolve the treacle of complexity that downs our best theories.

Jon Awbrey

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http://www.cs.bsu.edu/homepages/mighty/history.html
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