How do spectrum gaps relate to topological protected states? In particular, I want to understand what fundamental (mathematical) structure gives rise to topological mechanical metamaterials, or topological protected states in general. According to a recent PNAS paper "Topological mechanics of gyroscopic metamaterials", gapped spectrum of excitations plays an important role:

A vast range of mechanical structures, including bridges, covalent glasses, and conventional metamaterials, can be ultimately modeled as networks of masses connected by springs. Recent studies have revealed that despite its apparent simplicity, this minimal setup is sufficient to construct topologically protected mechanical states that mimic the properties of their quantum analogs. This follows from the fact that, irrespective of its classic or quantum nature, a periodic material with a gapped spectrum of excitations can display topological behavior as a result of the nontrivial topology of its band structure.

Can anyone give me an intuitive or mathematical explanation of why this is the case? I would appreciate if the explanation doesn't rely too much on jargons from quantum mechanics or solid state physics, as I am no expert in those fields and using the language of a specific type of physics could sometimes obscure the underlying mathematical structure (but please feel free to give electronic or optical systems as examples). Thank you.
 A: First "topological protected states" are states of many bodies.
The gapped spectrum of excitations may have two interpretations:
(A) The band structure  has gaps. (In this case the "spectrum" is a single-particle spectrum)
(B) The excitations on top of many-body ground state have a gap. (In this case the "spectrum" is a many-particle spectrum)
For the case (A), the gapped spectrum (the gapped band structure) has nothing to do with "topological protected states", since the bands can be empty and there may not be any particles.
Only the free fermion systems with filled bands may correspond to "topological protected states". The free fermion systems with filled bands is a special case of case (B). 
For the case (B) the gapped many-body ground states may correspond to "topological protected states". For details, see What does it mean for a Hamiltonian or system to be gapped or gapless?
For the present question, one needs to clarify "topological protected state" of what particles? 
Is it the topological protected state of the atoms that form the material? In this case, the state is not even topological, since the state is not gapped.
Is it the topological protected state of the phonons with a gapped band structure? In this case the state is not even topological, since there is no phonons at T=0.
(There are many misuses of "topological" to describe things that are not topological.)
