The concept of t-bonded sets was briefly introduced by the second author in 1976 under the name of d-connected sets, though it has not received due consideration. This concept is a generalization of the concept of Delone (r, R)-systems. In light of the developments in the local theory for crystals that occurred since 1976 and demands in chemistry and crystallography, we believe the local theory for t-bonded sets deserves to be developed to describe materials whose atomic structures is multi-regular “microporous” point set. For a better description of such “microporous” structures it is worthwhile to take into consideration a parameter that represents atomic bonds within the matter. The overarching goal of this paper is to prove that analogous local conditions that guarantee that a Delone set is a regular (or multi-regular) system also guarantee that a t-bonded set is a regular (or multi-regular) t-bonded system.
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