Topological Structures on DMC Spaces †

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.

Continuity of channel parameters and operations under various DMC topologies

“…We have shown in [5] that DMC Moreover, for every 1 ≤ n ≤ m, there exists a canonical subspace of DMC [5]. Therefore, we can consider DMC…”

“…The strong topology is sequential, compactly generated, and T 4 [5]. On the other hand, if |X | ≥ 2, the strong topology is not first-countable anywhere [5], hence it is not metrizable.…”

“…Every natural topology is σ-compact, separable and path-connected [5]. On the other hand, if |X | ≥ 2, a Hausdorff natural topology is not Baire and it is not locally compact anywhere [5].…”

“…1 In Section II, we introduce the preliminaries for this paper. In Section III, we recall the main results of [5] that we need here. In Section IV, we introduce the channel parameters and operations that we investigate in this paper.…”

“…In [5], many topologies were constructed for the space of equivalent channels sharing a fixed input alphabet. In this paper, we study the continuity of many channel parameters and operations under these topologies.…”