Locally compact, ω1\omega_1-compact spaces

An $\omega_1$-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, $\omega_1$-compact space is $\sigma$-countably compact, i.e., the union of countably many countably compact spaces. These conditions involve very elementary properties.Comment: 21 pages, submitted, comments are welcom

Lossless Analog Compression

We establish the fundamental limits of lossless analog compression by considering the recovery of arbitrary m-dimensional real random vectors x from the noiseless linear measurements y=Ax with n x m measurement matrix A. Our theory is inspired by the groundbreaking work of Wu and Verdu (2010) on almost lossless analog compression, but applies to the nonasymptotic, i.e., fixed-m case, and considers zero error probability. Specifically, our achievability result states that, for almost all A, the random vector x can be recovered with zero error probability provided that n > K(x), where K(x) is given by the infimum of the lower modified Minkowski dimension over all support sets U of x. We then particularize this achievability result to the class of s-rectifiable random vectors as introduced in Koliander et al. (2016); these are random vectors of absolutely continuous distribution---with respect to the s-dimensional Hausdorff measure---supported on countable unions of s-dimensional differentiable submanifolds of the m-dimensional real coordinate space. Countable unions of differentiable submanifolds include essentially all signal models used in the compressed sensing literature. Specifically, we prove that, for almost all A, s-rectifiable random vectors x can be recovered with zero error probability from n>s linear measurements. This threshold is, however, found not to be tight as exemplified by the construction of an s-rectifiable random vector that can be recovered with zero error probability from n<s linear measurements. This leads us to the introduction of the new class of s-analytic random vectors, which admit a strong converse in the sense of n greater than or equal to s being necessary for recovery with probability of error smaller than one. The central conceptual tools in the development of our theory are geometric measure theory and the theory of real analytic functions

On the commutativity of the powerspace constructions

We investigate powerspace constructions on topological spaces, with a particular focus on the category of quasi-Polish spaces. We show that the upper and lower powerspaces commute on all quasi-Polish spaces, and show more generally that this commutativity is equivalent to the topological property of consonance. We then investigate powerspace constructions on the open set lattices of quasi-Polish spaces, and provide a complete characterization of how the upper and lower powerspaces distribute over the open set lattice construction

Reduced Coproducts of Compact Hausdorff Spaces

By analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the reduced coproduct , which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the ultracoproduct can be obtained from the usual topological ultraproduct via a compactification process in the style of Wallman and Frink. We prove theorems dealing with the topological structure of reduced coproducts (especially ultracoproducts) and show in addition how one may use this construction to gain information about the category of compact Hausdorff spaces