Mathematics

Using the classical technique of condensation of singularities, we prove that, for every zero-dimensional, complete separable metric space G , there exists a Suslinian, chainable metric continuum whose set of end points is homeomorphic to G . This answers a question posed by R. Adikari and W. Lewis in [Houston J. Math. 45 (2019), no. 2, pp. 609--624].

In nonstandard analysis, Fehrele's principle is a beautiful criterion for a set to be internal, stating that every galactic halic set is internal. In this note, we use this principle to prove some well-known results in topology, including slight generalisations of the Moore-Osgood theorem and Dini's theorem.

Braided vector fields on spatial subdomains homeomorphic to the cylinder play a crucial role in applications such as solar and plasma physics, relativistic astrophysics, fluid and vortex dynamics, elasticity, and bio-elasticity. Often the vector field's topology -- the entanglement of its field lines -- is non-trivial, and can play a significant role in the vector field's evolution. We present a complete topological characterisation of such vector fields (up to isotopy) using a quantity called field line winding. This measures the entanglement of each field line with all other field lines of the vector field, and may be defined for an arbitrary tubular subdomain by prescribing a minimally distorted coordinate system. We propose how to define such coordinates, and prove that the resulting field line winding distribution uniquely classifies the topology of a braided vector field. The field line winding is similar to the field line helicity considered previously for magnetic (solenoidal) fields, but is a more fundamental measure of the field line topology because it does not conflate linking information with field strength.

A standard problem in applied topology is how to discover topological invariants of data from a noisy point cloud that approximates it. We consider the case where a sample is drawn from a properly embedded C1-submanifold without boundary in a Euclidean space. We show that we can deformation retract the union of ellipsoids, centered at sample points and stretching in the tangent directions, to the manifold. Hence the homotopy type, and therefore also the homology type, of the manifold is the same as that of the nerve complex of the cover by ellipsoids. By thickening sample points to ellipsoids rather than balls, our results require a smaller sample density than comparable results in the literature. They also advocate using elongated shapes in the construction of barcodes in persistent homology.

We provide a representation of the homomorphisms U⟶R , where U is the lattice of all uniformly continuous on the line. The resulting picture is sharp enough to describe the fine topological structure of the space of such homomorphisms.

We construct, using mild combinatorial hypotheses, a real Menger set that is not Scheepers, and two real sets that are Menger in all finite powers, with a non-Menger product. By a forcing-theoretic argument, we show that the same holds in the Blass--Shelah model for arbitrary values of the ultrafilter and dominating number.

We first introduce and study two new classes of subsets in T 0 spaces - ω -Rudin sets and ω -well-filtered determined sets lying between the class of all closures of countable directed subsets and that of irreducible closed subsets, and two new types of spaces - ω - d spaces and ω -well-filtered spaces. We prove that an ω -well-filtered T 0 space is locally compact iff it is core compact. One immediate corollary is that every core compact well-filtered space is sober, answering Jia-Jung problem with a new method. We also prove that all irreducible closed subsets in a first countable ω -well-filtered T 0 space are directed. Therefore, a first countable T 0 space X is sober iff X is well-filtered iff X is an ω -well-filtered d -space. Using ω -well-filtered determined sets, we present a direct construction of the ω -well-filtered reflections of T 0 spaces, and show that products of ω -well-filtered spaces are ω -well-filtered.

In this paper, we study some new fixed point results for self maps defined on partial metric type spaces. In particular, we give common fixed point theorems in the same setting. Some examples are given which illustrate the results.

In this paper, we examine some properties of the fixed point set of a digitally continuous function. The digital setting requires new methods that are not analogous to those of classical topological fixed point theory, and we obtain results that often differ greatly from standard results in classical topology. We introduce several measures related to fixed points for continuous self-maps on digital images, and study their properties. Perhaps the most important of these is the fixed point spectrum F(X) of a digital image: that is, the set of all numbers that can appear as the number of fixed points for some continuous self-map. We give a complete computation of F( C n ) where C n is the digital cycle of n points. For other digital images, we show that, if X has at least 4 points, then F(X) always contains the numbers 0, 1, 2, 3, and the cardinality of X . We give several examples, including C n , in which F(X) does not equal {0,1,…,#X} . We examine how fixed point sets are affected by rigidity, retraction, deformation retraction, and the formation of wedges and Cartesian products. We also study how fixed point sets in digital images can be arranged; e.g., in some cases the fixed point set is always connected.

We force a classification of all the Abelian groups of cardinality at most 2 c that admit a countably compact group with a non-trivial convergent sequence. In particular, we answer (consistently) Question 24 of Dikranjan and Shakhmatov for cardinality at most 2 c , by showing that if a non-torsion Abelian group of size at most 2 c admits a countably compact Hausdorff group topology, then it admits a countably compact Hausdorff group topology with non-trivial convergent sequences.