Graph topology and gap topology for unstable systems

A reformation is provided of the graph topology and the gap topology for a general setting (including lumped linear time-invariant systems and distributed linear time-invariant systems) in the frequency domain. Some essential properties and their comparisons are clearly presented in the reformulation. It is shown that the gap topology is suitable for general systems rather than square systems with unity feedback. It is shown that whenever an unstable plant can be stabilized by feedback, it is a closed operator, mapping a subspace of the input space to the output space. Hence, the gap topology can always be applied whenever the unstable plant can be stabilized. The graph topology and the gap topology are suitable for different subsets of systems and have many similar characteristics. If one confines them to the same subset, they will be identical. The definitions of the graph metric and the gap metric are discussed