Let C be a closed convex cone in a Banach ideal space X on a measurable space with a σ-finite measure. We prove that conditions C ∩ X+ = {0} and C ⊃ −X+ imply the existence of a strictly positive continuous functional on X , whose restriction to C is non-positive. Let (Ω,F ) be a measurable space, which is complete with respect to a measure (that is, a countably-additive function) μ : F 7→ [0,∞...

We show that every completely regular frame has a P -frame re‡ection. The proof is straightforward in the case of a Lindelöf frame, but more complicated in the general case. The chief obstacle to a simple proof is the important fact that a quotient of a P -frame need not be a P -frame, and we give an example of this. Our proof of the existence of the P -frame re‡ection in the general case is it...

we study umbilic (space-like) submanifolds of pseudo-riemannian space forms, then define totally semi-umbilic space-like submanifold of pseudo euclidean space and relate this notion to umbilicity. finally we give characterization of total semi-umbilicity for space-like submanifolds contained in pseudo sphere or pseudo hyperbolic space or the light cone.a pseudo-riemannian submanifold m in (a ps...

Given a map T : X → X on a set X we examine under what conditions there is a separable metrizable or an hereditarily Lindelöf or a Lindelöf topology on X with respect to which T is a continuous map. For separable metrizable and hereditarily Lindelöf, it turns out that there is a such a topology precisely when the cardinality ofX is no greater than the cardinality of the continuum. We go onto pr...

Given a map T : X → X on a set X we examine under what conditions there is a separable metrizable or an hereditarily Lindelöf or a Lindelöf topology on X with respect to which T is a continuous map. For separable metrizable and hereditarily Lindelöf, it turns out that there is such a topology precisely when the cardinality of X is no greater than c, the cardinality of the continuum. We go on to...

The original aim of this paper is to construct a graph associated to a vector space. By inspiration of the classical definition for the Cayley graph related to a group we define Cayley graph of a vector space. The vector space Cayley graph ${rm Cay(mathcal{V},S)}$ is a graph with the vertex set the whole vectors of the vector space $mathcal{V}$ and two vectors $v_1,v_2$ join by an edge whenever...

A subset A of a space X is called regular open if A = IntA, and regular closed if X\A is regular open, or equivalently, if A= IntA. A is called semiopen [16] (resp., preopen [17], semi-preopen [3], b-open [4]) ifA⊂ IntA (resp.,A⊂ IntA,A⊂ IntA ,A⊂ IntA∪ IntA). The concept of a preopen set was introduced in [6] where the term locally dense was used and the concept of a semi-preopen set was introd...