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Some necessary and sufficient conditions allowing a previously unknown space to be explored through scanning operators are reexamined with respect to measure theory. Some generalized concepts 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 non‐wellfounded properties of the empty set is demonstrated. This lattice provides a substratum with both discrete and continuous properties from which existence of physical universes can be proved, up to the function of conscious perception. Space‐time emerges as an ordered sequence of mappings of closed 3D Poincaré sections of a topological four‐space provided by the lattice, and the function of conscious perception is founded on the same properties. Self‐evaluation of a system is possible against indecidability barriers through anticipatory mental imaging occurring in biological brain systems; then our embedding universe should be in principle accessible to knowledge. 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 presented here 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 detail in Parts 2 and 3 of the study.

The non‐well‐founded properties of the empty set provide existence to abstract topological spaces, in which intersections of subspaces with non‐equal dimensions give topologically closed structures. This let space‐time emerge as an ordered sequence of sections upon a defined combination rule, in which interactivity between observable structures is allowed and conditions for conscious perception phenomena are fulfilled. Conditions for evolution and rules for optimal evolution of ecosystems also infer as corollaries. A physical universe and life constitute one single self‐organized and self managed system where life appears as the physical‐like realization of conditions of functionality of the embedding mathematical spaces. The system includes self‐ethical and moral guidelines which should inspire human behavior on Planet Earth.

An abstract lattice of empty set cells is shown to be able to account for a primary substrate in a physical space. Space‐time is represented by ordered sequences of topologically closed Poincaré sections of this primary space. These mappings are constrained to provide homeomorphic structures serving as frames of reference in order to account for the successive positions of any objects present in the system. Mappings from one section to the next involve morphisms of the general structures, representing a continuous reference frame, and morphisms of objects present in the various parts of this structure. The combination of these morphisms provides space‐time with the features of a non‐linear generalized convolution. Discrete properties of the lattice allow the prediction of scales at which microscopic to cosmic structures should occur. Deformations of primary cells by exchange of empty set cells allow a cell to be mapped into an image cell in the next section as far as the mapped cells remain homeomorphic. However, if a deformation involves a fractal transformation to objects, there occurs a change in the dimension of the cell and the homeomorphism is not conserved. Then, the fractal kernel stands for a “particle” and the reduction of its volume (together with an increase in its area up to infinity) is compensated by morphic changes of a finite number of surrounding cells. Quanta of distances and quanta of fractality are demonstrated. The interactions of a moving particle‐like deformation with the surrounding lattice involves a fractal decomposition process, which supports the existence and properties of previously postulated inerton clouds as associated to particles. Experimental evidence of the existence of inertons is reviewed and further possibilities of experimental proofs proposed.

The distribution of the deformations of elementary cells is studied in an abstract lattice constructed from the existence of the empty set. One combination rule determining oriented sequences with continuity of set‐distance function in such spaces provides a particular kind of space‐time‐like structure which favors the aggregation of such deformations into fractal forms standing for massive objects. A correlative dilatation of space appears outside the aggregates. At large scale, this dilatation results in an apparent expansion, while at submicroscopic scale the families of fractal deformations give rise to families of particle‐like structure. The theory predicts the existence of classes of spin, charges and magnetic properties, while quantum properties associated with mass have previously been shown to determine the inert mass and the gravitational effects. When applied to our observable space‐time, the model would provide the justifications for the existence of the creation of mass in a specified kind of void, and the fractal properties of the embedding lattice extend the phenomenon to formal justifications of big‐bang‐like events without any need for supply of an extemporaneous energy.