The graph topology on nth-order systems is quotient Euclidean

It is shown that on the set of m-input p-output minimal nth-order state-space systems the graph topology and the induced Euclidean quotient topology are identified. The author considers the set L_n^p×m of m -input p-output nth-order minimal state-space systems. The author presents three lemmas and a corollary from which a theorem is proved stating that the graph topology and the quotient Euclidean topology are identical on a quotient space L_n ^p×m/~. Since the graph topology is constructed to be weak, and the quotient Euclidean topology is intuitively strong, this result is unexpected