Continuity of Channel Parameters and Operations under Various DMC Topologies †

We study the continuity of many channel parameters and operations under various topologies on the space of equivalent discrete memoryless channels (DMC). We show that mutual information, channel capacity, Bhattacharyya parameter, probability of error of a fixed code and optimal probability of error for a given code rate and block length are continuous under various DMC topologies. We also show that channel operations such as sums, products, interpolations and Arıkan-style transformations are continuous.