The article of Topology-based representation (page no. 13, line 5) says that, topology-based representation is invariant to certain changes in the environment. That means the trajectory generated in topology-based space will remain valid even if there are certain changes in the environment. But how is this possible? Is there any simple example to understand this concept?
The concept you might be missing is homeomorphism. The donut=coffee cup visual example at that link might help. Of all the ways you can change the environment (move the obstacles) there is a subset of changes you can make that change the location of the obstacles but keeps the topology the same. Because the topology is the same, the topology based space trajectory will remain valid, even though the obstacles have moved. (Again, depending on the topology, only a subset of all possible changes will keep the same topology.)