In this brief note, we prove that every space is a sequence-covering image of a topological sum of convergent sequences. As the application of this result, sequence-covering images of locally compact metric spaces (or, locally separable metric spaces, metric spaces) and sequentially-quotient images of a locally compact metric spaces (or, locally separable metric spaces, metric spaces) are equiv...

The weak topology of a locally convex space (lcs) E is denoted by w. In this paper we undertake systematic study those lcs such that (E, w) (linearly) Eberlein–Grothendieck (see Definitions 1.2 and 3.1). following results obtained in our play key role: for every barrelled E, the (linearly Eberlein–Grothendieck) if only metrizable (E normable, respectively). main applications concern to continuo...

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...

— We present a constructive proof of Alexandrov’s theorem on the existence of a convex polytope with a given metric on the boundary. The polytope is obtained by deforming certain generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes and discover a connection with weighted Delaunay triangulations of polyhedral surfaces. The existence of the defo...

In this paper we introduce the concept of topological number for locally convex topological spaces and prove some of its properties. It gives some criterions to study locally convex topological spaces in a discrete approach.

Some results are obtained on finite unions ofD-spaces. It is proved that if a space is the union of finitely many locally compact Dsubspaces, then it is aD-space. It follows that a space is aD-space if it is the union of finitely many locally compact submetacompact subspaces. And a space is a D-space if it is the union of a D-subspace with a locally compact D-subspace. This partially answers on...