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Necessary and sufficient conditions allowing a previously unknown space to beexplored through scanning operators are reexamined with respect to measuretheory. Generalized conceptions of distances and dimensionality evaluation areproposed, together with their conditions of validity and range of applicationto topological spaces. The existence of a Boolean lattice with fractalproperties originating from nonwellfounded properties of the empty set isdemonstrated. This lattice provides a substrate with both discrete andcontinuous properties, from which existence of physical universes can beproved, up to the function of conscious perception. Spacetime emerges as anordered sequence of mappings of closed 3-D Ponicare sections of a topological4-space provided by the lattice. The possibility of existence of spaces withfuzzy dimension or with adjoined parts with decreasing dimensions is raised,together with possible tools for their study. The work provides theintroductory foundations supporting a new theory of space whose physicalpredictions (suppressing the opposition of quantum and relativistic approaches)and experimental proofs are presented in details in Parts 2 and 3 of the study.

Scanning the structure of ill‐known spaces: Part 1. Founding principles about mathematical constitution of space