Topology-dependent modifiers perform operations on explicit, topological sub-object selections. The Edit Mesh and Mesh Select modifiers are examples of modifiers that perform operations or selections on explicit vertex or face numbers. When these modifiers are present in the stack, you can adversely affect their results if you visit previous stack operations and change the topology (the number and order of faces and vertices) being passed to them. When you do this, a topology-dependence warning alerts you to the situation.