  Necessary and sufficient conditions allowing a previously unknown space to be
explored through scanning operators are reexamined with respect to measure
theory. Generalized conceptions of distances and dimensionality evaluation are
proposed, together with their conditions of validity and range of application
to topological spaces. The existence of a Boolean lattice with fractal
properties originating from nonwellfounded properties of the empty set is
demonstrated. This lattice provides a substrate with both discrete and
continuous properties, from which existence of physical universes can be
proved, up to the function of conscious perception. Spacetime emerges as an
ordered sequence of mappings of closed 3-D Ponicare sections of a topological
4-space provided by the lattice. The possibility of existence of spaces with
fuzzy dimension or with adjoined parts with decreasing dimensions is raised,
together with possible tools for their study. The work provides the
introductory foundations supporting a new theory of space whose physical
predictions (suppressing the opposition of quantum and relativistic approaches)
and experimental proofs are presented in details in Parts 2 and 3 of the study.
