Let PR(X) denote the hyperspace of nonempty finite subsets a topological spaceX with Pixley-Roy topology. In this paper, by introducing closed-miss-finite networks and using principle ultrafilters, we proved that following statements are equivalent for space X: (1) is weakly Rothberger; (2) X satisfies S1(?rcf ,?wrcf ); (3) separable ? {x} S1(?cf ,?wcf ) each x X; (4) principal ultrafilter F [x...

Let M be either a topological manifold, a Hilbert cube manifold, or a Menger manifold and let D be an arbitrary countable dense subset of M . Consider the topological group H(M,D) which consists of all autohomeomorphisms of M that map D onto itself equipped with the compact-open topology. We present a complete solution to the topological classification problem for H(M,D) as follows. If M is a o...

The local structure of homogeneous continua (curves) is studied. Components of open subsets of each homogeneous curve which is not a solenoid have the disjoint arcs property. If the curve is aposyndetic, then the components are nonplanar. A new characterization of solenoids is formulated: a continuum is a solenoid if and only if it is homogeneous, contains no terminal nontrivial subcontinua and...

Given a graph G = (V; E) and two vertices s; t 2 V , s 6 = t, the Menger problem is to nd a maximum number of disjoint paths connecting s and t. Depending on whether the input graph is directed or not, and what kind of disjointness criterion is demanded, this general formulation is specialized to the directed or undirected vertex, and the edge or arc disjoint Menger problem, respectively. For p...

Given a graph G V E and two vertices s t V s t the Menger problem is to nd a maximum number of disjoint paths connecting s and t Depending on whether the input graph is directed or not and what kind of disjointness criterion is demanded this general formulation is specialized to the directed or undirected vertex and the edge or arc disjoint Menger problem respectively For planar graphs the edge...

The additivity number of a topological property (relative to a given space) is the minimal number of subspaces with this property whose union does not have the property. The most well-known case is where this number is greater than א0, i.e. the property is σ-additive. We give a rather complete survey of the known results about the additivity numbers of a variety of topological covering properti...

In [16] K. Menger proposed the probabilistic concept of distance by replacing the number d(p, q), as the distance between points p, q, by a distribution function Fp,q. This idea led to development of probabilistic analysis [3], [11] [18]. In this paper, generalized probabilistic 2-normed spaces are studied and topological properties of these spaces are given. As an example, a space of random va...