Title: 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 $\sigma$-compact, separable and path-connected. On the
other hand, if $|X|\geq 2$, a Hausdorff natural topology is not Baire and it is
not locally compact anywhere. This implies that no natural topology can be
completely metrized if $|X|\geq 2$. 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 show that in the strong topology, a subspace is
compact if and only if it is rank-bounded and strongly-closed. 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. We show that the weak-* topology is exactly the
same as the noisiness topology and hence it is natural. We prove that if
$|X|\geq 2$, the total variation topology is not natural nor Baire, hence it is
not completely metrizable. Moreover, it is not locally compact anywhere.
Finally, we show that the Borel $\sigma$-algebra is the same for all Hausdorff
natural topologies.