Abstract:
The research of the macro-topology between large-scale software and AS-level Internet has important significances in the further comprehension and application for their structures. According to the complex network characteristic which is reflected by the macro-topology between largcscale software and A}level Internet, their structures were converted to network topology, analyzed and compared from connectivity, degree distribution characteristic, small-world characteristic and hierarchy, using the methods of complex network metric and analysis, then some macro-topology similarities and differences between large-scale software and AS-level Internet were discovered. Additionally, the reasons of them were discussed.

Abstract:
An example of a D-metric space is given, in which D-metric convergence does not define a topology and in which a convergent sequence can have infinitely many limits. Certain methods for constructing D-metric spaces from a given metric space are developed and are used in constructing (1) an example of a D-metric space in which D-metric convergence defines a topology which is T1 but not Hausdorff, and (2) an example of a D-metric space in which D-metric convergence defines a metrizable topology but the D-metric is not continuous even in a single variable.

Abstract:
In this paper, we prove that the topology induced by algebraic cone metric coincides with the topology induced by the metric obtained via a nonlinear scalarization function, i.e. any algebraic cone metric space is metrizable. Furthermore, the notion of algebraic cone normed space is introduced and also normability of the topology of this space is proved.

In this paper, we give a comment on the dislocated-neighbourhood systems due to Hitzler and Seda [1]. Also, we recover the open sets of the dislocated topology.

Abstract:
The necessity of a theory of General Topology and, most of all, of Algebraic Topology on locally finite metric spaces comes from many areas of research in both Applied and Pure Mathematics: Molecular Biology, Mathematical Chemistry, Computer Science, Topological Graph Theory and Metric Geometry. In this paper we propose the basic notions of such a theory and some applications: we replace the classical notions of continuous function, homeomorphism and homotopic equivalence with the notions of NPP-function, NPP-local-isomorphism and NPP-homotopy (NPP stands for Nearest Point Preserving); we also introduce the notion of NPP-isomorphism. We construct three invariants under NPP-isomorphisms and, in particular, we define the fundamental group of a locally finite metric space. As first applications, we propose the following: motivated by the longstanding question whether there is a purely metric condition which extends the notion of amenability of a group to any metric space, we propose the property SN (Small Neighborhood); motivated by some applicative problems in Computer Science, we prove the analog of the Jordan curve theorem in $\mathbb Z^2$; motivated by a question asked during a lecture at Lausanne, we extend to any locally finite metric space a recent inequality of P.N.Jolissaint and Valette regarding the $\ell_p$-distortion.

Abstract:
Let $\calA_+$ denote the set of Laplace transforms of complex Borel measures $\mu$ on $[0,+\infty)$ such that $\mu$ does not have a singular non-atomic part. In \cite{BalSas}, an extension of the classical $\nu$-metric of Vinnicombe was given, which allowed one to address robust stabilization problems for unstable plants over $\calA_+$. In this article, we show that this new $\nu$-metric gives a topology on unstable plants which coincides with the classical gap topology for unstable plants over $\calA_+$ with a single input and a single output.

Abstract:
There exists a completely metrizable bounded metrizable space $X$ with compatible metrics $d,d'$ so that the hyperspace $CL(X)$ of nonempty closed subsets of $X$ endowed with the Hausdorff metric $H_d$, $H_{d'}$, resp. is $\alpha$-favorable, $\beta$-favorable, resp. in the strong Choquet game. In particular, there exists a completely metrizable bounded metric space $(X,d)$ such that $(CL(X),H_d)$ is not completely metrizable.

Abstract:
We prove that, for any topological space $X$ and any metric space $(Y,d)$, the fine topology on the space of continuous functions from $X$ into $Y$ is independent of the metric $d$.

Abstract:
A contact pair on a manifold always admits an associated metric for which the two characteristic contact foliations are orthogonal. We show that all these metrics have the same volume element. We also prove that the leaves of the characteristic foliations are minimal with respect to these metrics. We give an example where these leaves are not totally geodesic submanifolds.

Abstract:
We present an operator space version of Rieffel's theorem on the agreement of the metric topology, on a subset of the Banach space dual of a normed space, from a seminorm with the weak*-topology. As an application we obtain a necessary and sufficient condition for the matrix metric from an unbounded Fredholm module to give the BW-topology on the matrix state space of the $C^*$-algebra. Motivated by recent results we formulate a non-commutative Lipschitz seminorm on a matrix order unit space and characterize those matrix Lipschitz seminorms whose matrix metric topology coincides with the BW-topology on the matrix state space.