DMC空间的拓扑结构
Topological Structures on DMC spaces
摘要：
如果两个通道彼此降级，则称它们是等效的。具有输入字母表$X$和输出字母表$Y$的等效信道的空间可以通过等价关系自然地被赋予欧几里德拓扑的商。具有固定输入字母表$X$和任意但有限的输出字母表的等价通道空间上的拓扑称为自然的当且仅当它在共享相同输出字母表的等价通道的子空间上导出商拓扑。我们证明了每个自然拓扑都是$σ$ -紧的、可分的和路连通的。另一方面，如果$|X|\geq 2$，则Hausdorff自然拓扑不是Baire，并且它在任何地方都不是局部紧的。这意味着，如果$|X|\geq 2$。最好的自然拓扑，我们称之为强拓扑，证明了它是紧生成的、顺序的和$T_4$的。另一方面，强拓扑在任何地方都不是第一可数的，因此它是不可度量化的。我们证明了在强拓扑中，子空间是紧的当且仅当它是秩有界且强闭的。我们引入了一个度量距离的空间上的等效通道比较信道之间的噪声水平。诱导度量拓扑，我们称之为噪声拓扑，是自然的。我们还研究了拓扑结构，继承了从空间的Meta概率测度通过识别通道与他们的Blackwell措施。我们证明了弱-* 拓扑与噪声拓扑完全相同，因此它是自然的。证明了如果$|X|\geq 2$，则全变差拓扑不是自然的，也不是Baire的，因此它不是完全可度量化的.此外，它在任何地方都不是局部紧的。最后证明了Borel $σ$ -代数对所有Hausdorff自然拓扑都是相同的.
英文摘要：
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. 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 $σ$-algebra is the same for all Hausdorff natural topologies.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
