Mathematics

In this paper, we characterize infinite-dimensional manifolds modeled on absorbing sets in non-separable Hilbert spaces by using the discrete cells property, which is a general position property. Moreover, we study the discrete (locally finite) approximation property, which is an extension of the discrete cells property.

In this paper, we give and prove two Chatterjea type fixed point theorems on partial b -metric space. We propose an extension to the Banach contraction principle on partial b -metric space which was already presented by Shukla and also study some related results on the completion of a partial metric type space. In particular, we prove a joint Chatterjea-Kannan fixed point theorem. We verify the T -stability of Picard's iteration and conjecture the P property for such maps. We also give examples to illustrate our results.

We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We prove that each Hausdorff topological space can be embedded as a closed subspace into an H-closed topological space. However, the semigroup of ω×ω -matrix units cannot be embedded into a topological semigroup which is a weakly H-closed topological space. We show that each Hausdorff topological space is a closed subspace of some ω -bounded pracompact topological space and describe open dense subspaces of countably pracompact topological spaces. Also, we construct a pseudocompact topological semigroup which contains the bicyclic monoid as a closed subsemigroup, providing a positive solution of a problem posed by Banakh, Dimitrova, and Gutik.

A set X endowed with a coarse structure is called ballean or coarse space. For a ballean (X,E) , we say that two subsets A , B of X are close (linked) if there exists an entourage E∈E such that A⊆E[B] , B⊆E[A] (either A,B are bounded or contain unbounded close subsets). We explore the following general question: which information about a ballean is contained and can be extracted from the relations of closeness and linkness.

For a coarse space (X,E) , X ♯ denotes the set of all unbounded ultrafilters on X endowed with the parallelity relation: p||q if there exists E∈E such that E[P]∈q for each P∈p . If (X,E) is finitary then there exists a group G of permutations of X such that the coarse structure E has the base {{(x,gx):x∈X , g∈F}:F∈[G ] <ω , id∈F}. We survey and analyze interplays between (X,E) , X ♯ and the dynamical system (G, X ♯ ) .

A sequence ( a n ) in an Abelian group is called a T -sequence if there exists a Hausdorff group topology on G in which ( a n ) converges to 0 . For a T -sequence ( a n ) , τ ( a n ) denotes the strongest group topology on G in which ( a n ) converges to 0 . The ideal I ( a n ) of all precompact subsets of (G, τ ( a n ) ) defines a coarse structure on G with base of entourages {(x,y):x−y∈P} , P∈ I ( a n ) . We prove that asdim (G, I ( a n ) )=∞ for every non-trivial T -sequence ( a n ) on G , and the coarse group (G, I ( a n ) ) has 1 end provided that ( a n ) generates G . The keypart play asymorphic copies of the Hamming space in (G, I ( a n ) ) .

In this article, we investigate some properties of the coincidence point set of digitally continuous maps. Following the Rosenfeld graphical model which seems more combinatorial than topological, we expect to achieve results that might not be analogous to the classical topological fixed point theory. We also introduce and study some topological invariants related to the coincidence and common fixed point sets for continuous maps on a digital image. Moreover, we study how these coincidence point sets are affected by rigidity and deformation retraction. Lastly, we present briefly a concept of divergence degree of a point in a digital image.

The aim of this paper is to generalize some of the properties and results regarding both the coincidence point set and the common fixed point set of any two digitally continuous maps to the case of several (more than two) digitally continuous mappings. Moreover, we study how rigidity may affect these coincidence and homotopy coincidence point sets. Also, we investigate whether an established result by Staecker in Nielsen classical topology regarding the coincidence set for many maps still remains valid in the digital topological setting.

For a T 0 space X , let $\mk (X)$ be the poset of all compact saturated sets of X with the reverse inclusion order. The space X is said to have property Q if for any $K_1, K_2\in \mk (X)$, K 2 ≪ K 1 in $\mk (X)$ if{}f $K_2\subseteq \ii~\!K_1$. In this paper, we give several connections among the well-filteredness of X , the sobriety of X , the local compactness of X , the core compactness of X , the property Q of X , the coincidence of the upper Vietoris topology and Scott topology on $\mk (X)$, and the continuity of $x\mapsto\ua x : X \longrightarrow \Sigma~\!\! \mk (X)$ (where $\Sigma~\!\! \mk (X)$ is the Scott space of $\mk (X)$). It is shown that for a well-filtered space X for which its Smyth power space P S (X) is first-countable, the following three properties are equivalent: the local compactness of X , the core compactness of X and the continuity of $\mk (X)$. It is also proved that for a first-countable T 0 space X in which the set of minimal elements of K is countable for any compact saturated subset K of X , the Smyth power space P S (X) is first-countable. For the Alexandroff double circle Y , which is Hausdorff and first-countable, we show that its Smyth power space P S (Y) is not first-countable.

We gather in this note results and examples about collared or non-collared boundaries of non-metrisable manifolds. Almost everything is well known but a bit scattered in the literature, and some of it is apparently not published at all.