Interpretations of the gap topology: A survey

Publisher Summary This chapter presents a survey of some interpretations of the gap topology. The gap topology is defined by the gaps between the graphs of the transfer functions as linear operators from the input space to the output space. The gap topology on linear systems is usually defined as the topology induced by the gaps between the graphs of the transfer functions. It is natural also to consider the gaps between the solution sets. The topology on the whole line is strictly weaker, and the behavior topology on the right half line is uncomparable to the graph topology. Instead of describing a system in the frequency domain by its transfer function, one can also identify it with the associated Hermann–Martin mapping to a Grassmannian manifold. Continuity of interconnection of linear systems was the main reason to introduce the graph topology. Interconnection is most easily modeled simply as intersection of the associated behaviors.

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