We are interested in the regularity of centered Gaussian processes (Z x (ω)) x∈M indexed by compact metric spaces (M, ρ). It is shown that the almost everywhere Besov space regularity of such a process is (almost) equivalent to the Besov regularity of the covariance K(x, y) = E(Z x Z y) under the assumption that (i) there is an underlying Dirichlet structure on M which determines the Besov spac...

We explore almost-normality in Isbell-Mr\'owka spaces and some related concepts. use forcing to provide an example of almost-normal not normal almost disjoint family, the concept semi-normality spaces, define strongly $(\aleph_0, <\mathfrak c)$-separated families prove generic existence completely separable c)$-almost assuming $\mathfrak s=\mathfrak c$ b=\mathfrak c$. also a Tychonoff pseudocom...

where f is a continuous function of compact support onGwith values in V . This is not quite a trivial matter. If V is assumed to be complete, a definition by Riemann sums works quite well, but this is an unnecessarily strong requirement. The natural condition on V seems to be that it be quasi-complete, a somewhat technical but nonetheless valuable condition. Continuous representations of a loca...

where f is a continuous function of compact support onGwith values in V . This is not quite a trivial matter. If V is assumed to be complete, a definition by Riemann sums works quite well, but this is an unnecessarily strong requirement. The natural condition on V seems to be that it be quasi-complete, a somewhat technical but nonetheless valuable condition. Continuous representations of a loca...

We present a measure-theoretic condition for a property to hold “almost everywhere” on an infinite-dimensional vector space, with particular emphasis on function spaces such as C and L. Like the concept of “Lebesgue almost every” on finite-dimensional spaces, our notion of “prevalence” is translation invariant. Instead of using a specific measure on the entire space, we define prevalence in ter...

We study the geometry of the leaf closure space of regular and singular Riemannian foliations. We give conditions which assure that this leaf space is a singular symplectic or Kähler space. In recent years physicists and mathematicians working on mathematical models of physical phenomena have realised that modelling based on geometric structures on now classical smooth manifolds is insufficient...

In this article, we present the concept of supra paracompact spaces and study its basic properties. We elucidate its relationship with supra compact spaces and prove that the property of being a supra paracompact space is weakly hereditary and topological properties. Also, we provide some examples to show some results concerning paracompactness on topology are invalid on supra topology. Finally...