Mathematics

We show that the escaping set for f(z)=exp(z)−1 is nowhere σ -complete. This establishes that the escaping endpoint set E ˙ (f) is a first category almost zero-dimensional space which is F σδ and nowhere G δσ . Only two other elementary spaces with those properties are known: Q ω and Erdős space E . Previous work has shown that E ˙ (f) is homeomorphic to neither of those.

Miller, Stibich and Moore (2010) developed a set-valued coarse invariant σ(X,ξ) of pointed metric spaces. DeLyser, LaBuz and Tobash (2013) provided a different way to construct σ(X,ξ) (as the set of all sequential ends). This paper provides yet another definition of σ(X,ξ) . To do this, we introduce a metric on the set S(X,ξ) of coarse maps (N,0)→(X,ξ) , and prove that σ(X,ξ) is equal to the set of coarsely connected components of S(X,ξ) . As a by-product, our reformulation trivialises some known theorems on σ(X,ξ) , including the functoriality and the coarse invariance.

Using the idea of strong uniform convergence on bornology, Caserta, Di Maio and Kočinac studied open covers and selection principles in the realm of metric spaces (associated with a bornology) and function spaces (w.r.t. the topology of strong uniform convergence). We primarily continue in the line initiated before and investigate the behaviour of various selection principles related to these classes of bornological covers. In the process we obtain implications among these selection principles resulting in Scheepers' like diagrams. We also introduce the notion of strong-$\mathfrac{B}$-Hurewicz property and investigate some of its consequences. Finally, in C(X) with respect to the topology $\tau_{\mathfrac{B}}^s$ of strong uniform convergence, important properties like countable T -tightness, Reznichenko property are characterized in terms of bornological covering properties of X .

We study the approximate fixed point property (AFPP) for continuous single-valued functions and for continuous multivalued functions in digital topology. We extend what is known about these notions and discuss errors that have appeared in the literature.

E.D. Tymchatyn constructed a hereditarily locally connected continuum which can be approximated by a sequence of mutually disjoint arcs. We show the example re-opens a conjecture of G.T. Seidler and H. Kato about continua which admit positive entropy homeomorphisms. We prove that every indecomposable semicontinuum can be approximated by a sequence of disjoint subcontinua, and no composant of an indecomposable continuum can be embedded into a Suslinian continuum. We also prove that if Y is a hereditarily unicoherent Suslinian continuum, then there exists ε>0 such that every two ε -dense subcontinua of Y intersect.

Arhangelskii's properties α 2 and α 4 defined for convergent sequences may be characterized in terms of Scheeper's selection principles. We generalize these results to hold for more general collections and consider these results in terms of selection games.

A Tychonoff space X is called ({\em sequentially}) {\em Ascoli} if every compact subset (resp. convergent sequence) of C k (X) is evenly continuous, where C k (X) denotes the space of all real-valued continuous functions on X endowed with the compact-open topology. Various properties of (sequentially) Ascoli spaces are studied, and we give several characterizations of sequentially Ascoli spaces. Strengthening a result of Arhangel'skii we show that a hereditary Ascoli space is Fréchet--Urysohn. A locally compact abelian group G with the Bohr topology is sequentially Ascoli iff G is compact. If X is totally countably compact or near sequentially compact then it is a sequentially Ascoli space. The product of a locally compact space and an Ascoli space is Ascoli. If additionally X is a μ -space, then X is locally compact iff the product of X with any Ascoli space is an Ascoli space. Extending one of the main results of [18] and [16] we show that C p (X) is sequentially Ascoli iff X has the property (κ) . We give a necessary condition on X for which the space C k (X) is sequentially Ascoli. For every metrizable abelian group Y , Y -Tychonoff space X , and nonzero countable ordinal α , the space B α (X,Y) of Baire- α functions from X to Y is κ -Fréchet--Urysohn and hence Ascoli.

A Tychonoff space X is called ({\em sequentially}) {\em Ascoli} if every compact subset (resp. convergent sequence) of C k (X) is equicontinuous, where C k (X) denotes the space of all real-valued continuous functions on X endowed with the compact-open topology. The classical Ascoli theorem states that each compact space is Ascoli. We show that a pseudocompact space X is Asoli iff it is sequentially Ascoli iff it is selectively ω -bounded.

Ge and Lin (2015) proved the existence and the uniqueness of p-Cauchy completions of partial metric spaces under symmetric denseness. They asked if every (non-empty) partial metric space X has a p-Cauchy completion X ¯ such that X is dense but not symmetrically dense in X ¯ . We construct asymmetric p-Cauchy completions for all non-empty partial metric spaces. This gives a positive answer to the question. We also provide a nonstandard construction of partial metric completions.

The omega limit sets plays a fundamental role to construct global attractors for topological semi-dynamical systems with continuous time or discrete time. Therefore, it is important to know when omega limit sets become nonempty compact sets. The purpose of this paper is to understand the mechanism under which a given net of subsets of topological spaces is compact in the asymptotic sense. For this purpose, we introduce the notion of asymptotic compactness for nets of subsets and study the connection with the compactness of the limit sets. In this paper, for a given net of nonempty subsets, we prove that the asymptotic compactness and the property that the limit set is a nonempty compact set to which the net converges from above are equivalent in uniformizable spaces. We also study the sequential version of the notion of asymptotic compactness by introducing the notion of sequentiality of directed sets.