The Spaces of Data, Information, and Knowledge
  We study the data space $D$ of any given data set $X$ and explain how
functions and relations are defined over $D$. From $D$ and for a specific
domain $\Delta$ we construct the information space $I$ of $X$ by interpreting
variables, functions, and explicit relations over $D$ in $\Delta$ and by
including other relations that $D$ implies under the interpretation in
$\Delta$. Then from $I$ we build up the knowledge space $K$ of $X$ as the
product of two spaces $K_T$ and $K_P$, where $K_T$ is obtained from $I$ by
using the induction principle to generalize propositional relations to
quantified relations, the deduction principle to generate new relations, and
standard mechanisms to validate relations and $K_P$ is the space of
specifications of methods with operational instructions which are valid in
$K_T$. Through our construction of the three topological spaces the following
key observation is made clear: the retrieval of information from the given data
set for $\Delta$ consists essentially in mining domain objects and relations,
and the discovery of knowledge from the retrieved information consists
essentially in applying the induction and deduction principles to generate
propositions, synthesizing and modeling the information to generate
specifications of methods with operational instructions, and validating the
propositions and specifications. Based on this observation, efficient
approaches may be designed to discover profound knowledge automatically from
simple data, as demonstrated by the result of our study in the case of
geometry.
