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Entropy 2018, 20(5), 343; https://doi.org/10.3390/e20050343

Topological Structures on DMC Spaces

École Polytechnique Fédérale de Lausanne, Route Cantonale, 1015 Lausanne, Switzerland
This paper is an extended version of our paper that is published in the International Symposium on Information Theory 2017 (ISIT 2017).
Received: 25 March 2018 / Revised: 19 April 2018 / Accepted: 27 April 2018 / Published: 4 May 2018
(This article belongs to the Section Information Theory, Probability and Statistics)
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Abstract

Two channels are said to be equivalent if they are degraded from each other. The space of equivalent channels with input alphabet X and output alphabet Y can be naturally endowed with the quotient of the Euclidean topology by the equivalence relation. A topology on the space of equivalent channels with fixed input alphabet X and arbitrary but finite output alphabet is said to be natural if and only if it induces the quotient topology on the subspaces of equivalent channels sharing the same output alphabet. We show that every natural topology is σ -compact, separable and path-connected. The finest natural topology, which we call the strong topology, is shown to be compactly generated, sequential and T 4 . On the other hand, the strong topology is not first-countable anywhere, hence it is not metrizable. We introduce a metric distance on the space of equivalent channels which compares the noise levels between channels. The induced metric topology, which we call the noisiness topology, is shown to be natural. We also study topologies that are inherited from the space of meta-probability measures by identifying channels with their Blackwell measures. View Full-Text
Keywords: discrete memoryless channels; topology; Blackwell measure; total-variation distance discrete memoryless channels; topology; Blackwell measure; total-variation distance
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Nasser, R. Topological Structures on DMC Spaces . Entropy 2018, 20, 343.

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