# Continuity of Channel Parameters and Operations under Various DMC Topologies

## TL;DR

This paper investigates the continuity properties of key channel parameters and operations in the space of equivalent discrete memoryless channels, providing foundational insights for information theory and channel coding.

## Contribution

It establishes the continuity of mutual information, capacity, Bhattacharyya parameter, and various channel operations under different topologies on DMCs, advancing theoretical understanding.

## Key findings

- Mutual information and capacity are continuous under various topologies.
- Channel operations like sums, products, and transformations are continuous.
- Key error probabilities are continuous functions of channel parameters.

## Abstract

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 blocklength, are continuous under various DMC topologies. We also show that channel operations such as sums, products, interpolations, and Ar{\i}kan-style transformations are continuous.