  We construct a theory to introduce the concept of topologically robust
exceptional points (EP). Starting from an ordered system with $N$ elements, we
find the necessary condition to have the highest order exceptional point,
namely $N^\text{th}$ order EP. Using symmetry considerations, we show an EP
associated with an order system is very sensitive to the disorder.
Specifically, if the EP associated with the ordered system occurs at the fixed
degree of non-Hermiticity $\gamma_{EP}$, the disordered system will not have EP
at the same $\gamma_{PT}$ which puts an obstacle in front of the observation
and applications of EPs. To overcome this challenge, by incorporating an
asymmetric coupling we propose a disordered system that has a robust EP which
is extended all over the space. While our approach can be easily realized in
electronic circuits and acoustics, we propose a simple experimentally feasible
photonic system to realize our robust EP. Our results will open a new direction
to search for topologically robust extended states (as opposed to topological
localized states) and find considerable applications in direct observation of
EPs, realizing topological sensors and designing robust devices for metrology.
