This paper is devoted to study the initial and final T-topogenous structures of T-topogenous spaces, defined by the
author in 2012. We show that all initial (final) lifts in the category T-TS of T-topogenous spaces and the initial (final) T-topogenous
structures exist. We introduce characterizations of the initial T-topogenous structures as special initial T-topogenous structures. The subspaces and product spaces of these T-topogenous spaces can be also characterized. Moreover, we show that the I-topology
associated to the initial T-topogenous structure of a family of T-topogenous spaces coincides with the initial I-topology of the fami
ly of I-topologies associated to these T-topogenous spaces.