  Disordered hyperuniform systems are exotic states of matter that completely
suppress large-scale density fluctuations like crystals, and yet possess no
Bragg peaks similar to liquids or glasses. Such systems have been discovered in
a variety of equilibrium and non-equilibrium physical and biological systems,
and are often endowed with novel physical properties. The mechanisms associated
with the emergence of disordered hyperuniformity in nonequilibrium systems, in
particular inherent structures are often not well understood, which we will
address from a topological perspective in this work. Specifically, we consider
a representative class of disordered inherent structures which are constructed
by continuously introducing randomly distributed topological defects
(dislocations and disclinations) often seen in colloidal systems and
atomic-scale two-dimensional materials. We demonstrate that these inherent
structures can be viewed as topological variants of ordered hyperuniform states
(such as crystals) linked through continuous topological transformation
pathways, which remarkably preserve hyperuniformity. Moreover, we develop a
continuum theory to demonstrate that the large-scale density fluctuations in
these inherent structures are mainly dominated by the elastic displacement
fields resulted from the topological defects, which at low defect
concentrations can be approximated as superposition of the displacement fields
associated with each individual defect (strain source). We rigorously
demonstrate that hyperuniformity is preserved as long as certain conditions
associated with the displacement fields are met. Our results also highlight the
importance of decoupling the positional degrees of freedom from the vibrational
degrees of freedom when looking for disordered hyperuniformity, since the
hyperuniformity property is often cloaked by thermal fluctuations.
