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Heinrich Tietze has shown that for a closed connected subset of euclidean space being convex is a local property. We generalize this to CAT(0)-spaces and locally compact CAT(\kappa) spaces. As an application we give a construction of…

In this note, we shall overview some results related to ultraparacompactness and ultranormality in the general topological and point-free contexts. This note contains some standard results and counterexamples along with some results which…

It is shown that the fixed point subalgebra of an EALA under a finite order automorphism (satisfying certain properties) is a sum of EALA's, an abelian subalgebra, and a subspace which is contained in the centralizer of the core.

A linear operator $T$ between two lattice-normed spaces is said to be $p$-compact if, for any $p$-bounded net $x_\alpha$, the net $Tx_\alpha$ has a $p$-convergent subnet. $p$-Compact operators generalize several known classes of operators…

Let $G$ be an infinite compact group. We prove that for every cardinal $\kappa$ between the density and the weight of $G$, there exists a dense subgroup of $G$ of density $\kappa$.

For a finite group scheme G over an algebraically closed field k of characteristic p>0 we study G-modules M, which are defined in terms of properties of their pull-backs along p-points of G. We show that the corresponding subcategories…

Subspace clustering is an important unsupervised clustering approach. It is based on the assumption that the high-dimensional data points are approximately distributed around several low-dimensional linear subspaces. The majority of the…

Complex networks can be used to represent complex systems which originate in the real world. Here we study a transformation of these complex networks into simplicial complexes, where cliques represent the simplices of the complex. We extend…

We organize a table of regular graphs with minimal diameters and minimal mean path lengths, large bisection widths and high degrees of symmetries, obtained by enumerations on supercomputers. These optimal graphs, many of which are newly…

Networks constitute efficient tools for assessing universal features of complex systems. In physical contexts, classical as well as quantum, networks are used to describe a wide range of phenomena, such as phase transitions, intricate…

Each straight infinite line defined by two vertices of a finite square point lattice contains (covers) these two points and a - possibly empty - subset of points that happen to be collinear to these. This work documents vertex subsets of…

The cross topology $\gamma$ on a product of topological spaces $X$ and $Y$ is the collection of all sets $G\subseteq X\times Y$ such that the intersection of $G$ with every vertical line and every horizontal line is an open subset of either…

The complement of a complexified real line arrangement is an affine surface. It is classically known that such a space has a handle decomposition up to $2$-handles. We will describe the handle decomposition induced from Lefschetz hyperplane…

For each countable ordinal $\alpha$, we introduce an ideal $conv_\alpha$ and use it to characterize the class of all compact countable spaces which are homeomorphic to the space $\omega^{\alpha}\cdot n+1$ with the order topology. The…

It is shown that for any Baire space $X$, linearly ordered compact spaces $Y_1,\dots, Y_n$, compact space $Y\subseteq Y_1\times\cdots \times Y_n$ such that for every parallelepiped $W\subseteq Y_1\times\cdots \times Y_n$ the set $Y\cap W$…

We provide a complete classification of the possible cofinal structures of the families of precompact (totally bounded) sets in general metric spaces, and compact sets in general complete metric spaces. Using this classification, we…

Connectedness, path connectedness, and uniform connectedness are well-known concepts. In the traditional presentation of these concepts there is a substantial difference between connectedness and the other two notions, namely connectedness…

The Reeb space, which generalizes the notion of a Reeb graph, is one of the few tools in topological data analysis and visualization suitable for the study of multivariate scientific datasets. First introduced by Edelsbrunner et al., it…

This work reviews several hierarchical measurements of the topology of complex networks and then applies feature selection concepts and methods in order to quantify the relative importance of each measurement with respect to the…

In this paper I survey some recent results on finite determination, convergence, and approximation of formal mappings between real submanifolds in complex spaces. A number of conjectures are also given.