Let A → B be an injective ring morphism. The The Traverso-Swan's seminormalization of A in B can be considered as the greatest subring C of B , such that the ring morphism A → C is spectrally bijective, with isomorphic residue field extensions. Assume now that A and B are Noetherian, Mori, integral domains, then T. Sugatani and K. Yoshida defined the t-closure of A in B as the greatest subring C of B such that the ring morphism A → C has isomorphic residue field extensions and is spectrally surjective. Following Swan's ideas about seminormalization, we have been able to delete the above hypothesis. We thus have obtained a general theory of t-closedness. Jnfra-integral morphisms in this work areanalogous with subintegral morphisms of Swan's work.