  One of the important characteristics of topological phases of matter is the
topology of the underlying manifold on which they are defined. In this paper,
we present the sensitivity of such phases of matter to the underlying topology,
by studying the phase transitions induced due to the change in the boundary
conditions. We claim that these phase transitions are accompanied by broken
symmetries in the excitation space and to gain further insight we analyze
various signatures like the ground state degeneracy, topological entanglement
entropy while introducing the open-loop operator whose expectation value
effectively captures the phase transition. Further, we extend the analysis to
an open quantum setup by defining effective collapse operators, the dynamics of
which cool the system to different topologically ordered steady states. We show
that the phase transition between such steady states is effectively captured by
the expectation value of the open-loop operator.
