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We study the continuity of many channel parameters and operations undervarious topologies on the space of equivalent discrete memoryless channels(DMC). We show that mutual information, channel capacity, Bhattacharyyaparameter, probability of error of a fixed code, and optimal probability oferror for a given code rate and blocklength, are continuous under various DMCtopologies. We also show that channel operations such as sums, products,interpolations, and Ar{\i}kan-style transformations are continuous.

Continuity of Channel Parameters and Operations under Various DMC Topologies