Let (Ω, d) be a metric space and let B be a partition of Ω. For every set B of B with positive and finite Hausdorff outer measure in its Hausdorff dimension, a coherent conditional measure of risk is defined as the Choquet integral with respect to Hausdorff outer measure. Two risks are defined to be s-independent if the atoms of the classes generated by their weak upper level sets are s-indepen...

We prove that the maximal Hausdorff compactification χf of a T2-compactifiable mapping f and the maximal Tychonoff compactification βf of a Tychonoff mapping f (see [P]) are perfect. This allows us to give a characterization of all perfect Hausdorff (respectively, all perfect Tychonoff) compactifications of a T2-compactifiable (respectively, of a Tychonoff) mapping, which is a generalization of...

We establish topological properties of the topological symmetric inverse semigroup of finite transformations I n λ of the rank 6 n. We show that the topological inverse semigroup I n λ is algebraically closed in the class of topological inverse semigroups. Many topologists established topological property of topological spaces of partial continuous maps PC (X, Y ) from a topological space X int...

Definition 1 (compare [2,5,8]) A topological space is called a KC-space provided that each compact set is closed. A topological space is called a U S-space provided that each convergent sequence has a unique limit. Remark 1 Each Hausdorff space (= T 2-space) is a KC-space, each KC-space is a U S-space and each U S-space is a T 1-space (that is, singletons are closed); and no converse implicatio...

For d ∈ {1,2,3}, let (B t ; t ≥ 0) be a d-dimensional standard Brownian motion. We study the d-Brownian span set Span(d) := {t − s;Bd s = B t for some 0 ≤ s ≤ t}. We prove that almost surely the random set Span(d) is σ -compact and dense in R+. In addition, we show that Span(1) = R+ almost surely; the Lebesgue measure of Span(2) is 0 almost surely and its Hausdorff dimension is 1 almost surely;...

In this paper we will introduce and develop a theory of adjunction spaces which allows the construction non-Hausdorff topological manifolds from standard Hausdorff ones. This is done by gluing along homeomorphic open submanifolds whilst leaving boundaries these regions unidentified. case that have boundaries, it shown violation occurs precisely at boundaries. We then use formalism to provide pa...

For a stationary set S ⊆ ω1 and a ladder system C over S, a new type of gaps called C-Hausdorff is introduced and investigated. We describe a forcing model of ZFC in which, for some stationary set S, for every ladder C over S, every gap contains a subgap that is CHausdorff. But for every ladder E over ω1 \ S there exists a gap with no subgap that is E-Hausdorff. A new type of chain condition, c...

In this paper we announce various new results about singularities in the mean curvature flow. Some results apply to any weak solution (i.e., any Brakke flow of integral varifolds.) Our strongest results, however, are for initially regular mean-convex hypersurfaces. (We say a hypersurface is mean-convex if it bounds a region such that the mean curvature with respect to the inward unit normal is ...