We calculate the density of the hyperspace of a metric space, endowed with the Hausdorff or the locally finite topology. To this end, we introduce suitable generalizations of the notions of totally bounded and compact metric space.

We show that the regularity of monomial ideals of K[x1, . . . , xn] (K being a field), whose associated prime ideals are totally ordered by inclusion is upper bounded by a linear function in n.

4 We prove that the weak k-linkage problem is polynomial for every fixed k for totally Φ5 decomposable digraphs, under appropriate hypothesis on Φ. We then apply this and recent results 6 by Fradkin and Seymour (on the weak k-linkage problem for digraphs of bounded independence 7 number or bounded cut-width) to get polynomial algorithms for some class of digraphs like quasi8 transitive digraphs...

We provide a method for factoring all bounded ratios of the form det A(I1|I ′ 1) det A(I2|I ′ 2)/ det A(J1|J ′ 1) det A(J2|J ′ 2) where A is a totally positive matrix, into a product of more elementary ratios each of which is bounded by 1, thus giving a new proof of Skandera’s result. The approach we use generalizes the one employed by Fallat et al. in their work on principal minors. We also ob...

For a left vector space V over a totally ordered division ring F, let Co(V ) denote the lattice of convex subsets of V . We prove that every lattice L can be embedded into Co(V ) for some left F-vector space V . Furthermore, if L is finite lower bounded, then V can be taken finite-dimensional, and L embeds into a finite lower bounded lattice of the form Co(V,Ω) = {X ∩Ω | X ∈ Co(V )}, for some f...

A subset E of a metric space (X, d) is totally bounded if and only if any sequence of points in E has a Cauchy subsequence. We call a sequence (xn) statistically quasiCauchy if st − limn→∞ d(xn+1, xn) = 0, and lacunary statistically quasi-Cauchy if Sθ − limn→∞ d(xn+1, xn) = 0.Weprove that a subset E of ametric space is totally bounded if and only if any sequence of points in E has a subsequence...

Definition 1.1 (Compact and Totally Bounded Sets). Let X be a metric space, and let E ⊆ X be given. (a) We say that E is compact if every open cover of E contains a finite subcover. That is, E is compact if whenever {Uα}α∈I is a collection of open sets whose union contains E, then there exist finitely many α1, . . . , αN such that E ⊆ Uα1 ∪ · · · ∪ UαN . (b) We say that E is sequentially compac...

We investigate the interrelations between labeled trees and ultrametric spaces generated by these trees. The trees, which generate complete ultrametrics, totally bounded discrete ones, are characterized up to isomorphism. As corollary, we obtain a characterization of generating compact ultrametrics ultrametrics. It is also shown that every space tree contains dense subspace.

It is shown that a bounded bi-infinite banded totally positive matrix A is boundedly invertible iff there is one and only one bounded sequence mapped by A to the sequence ((-)')■ The argument shows that such a matrix has a main diagonal, i.e., the inverse of A is the bounded pointwise limit of inverses of finite sections of A principal with respect to a particular diagonal; hence ((—)'+^4~ (j, ...