Strong Proximal Continuity and Convergence

In several situations the notion of uniform continuity can be strengthened to strong uniform continuity to produce interesting properties, especially in constrained problems. The same happens in the setting of proximity spaces. While a parallel theory for uniform and strong uniform convergence was recently developed, and a notion of proximal convergence is present in the literature, the notion of strong proximal convergence was never considered. In this paper, we propose several possible convergence notions, and we provide complete comparisons among these concepts and the notion of strong uniform convergence in uniform spaces. It is also shown that in particularly meaningful classes of functions these notions are equivalent and can be considered as natural definitions of strong proximal convergence. Finally we consider a function acting between two proximity spaces and we connect its continuity/strong continuity to convergence in the respective hyperspaces of a natural functor associated to the function itself

The Alexandroff Duplicate and its subspaces

[EN] We study some topological properties of the class of the Alexandroff duplicates and their subspaces. We give a characterization of metrizability and Lindel¨of properties of subspaces of the Alexandroff duplicate. This characterization clarifies the potential for finding Michael spaces among the subspaces of Alexandroff duplicates.Caserta, A.; Watson, S. (2007). The Alexandroff Duplicate and its subspaces. Applied General Topology. 8(2):187-205. doi:10.4995/agt.2007.1880.1872058

On resolutions of linearly ordered spaces

[EN] We define an extended notion of resolution of topologicalspaces, where the resolving maps are partial instead of total. To showthe usefulness of this notion, we give some examples and list severalproperties of resolutions by partial maps. In particular, we focus ourattention on order resolutions of linearly ordered sets. Let X be a setendowed with a Hausdorff topology τ and a (not necessarily related)linear order . A unification of X is a pair (Y, ı), where Y is a LOTSand ı : X →֒ Y is an injective, order-preserving and open-in-the-rangefunction. We exhibit a canonical unification (Y, ı) of (X,, τ ) such thatY is an order resolution of a GO-space (X,, τ ∗), whose topology τ ∗refines τ . We prove that (Y, ı) is the unique minimum unification ofX. Further, we explicitly describe the canonical unification of an orderresolution.Caserta, A.; Giarlotta, A.; Watson, S. (2006). On resolutions of linearly ordered spaces. Applied General Topology. 7(2):211-231. doi:10.4995/agt.2006.1925.SWORD21123172R. Engelking, General Topology (Heldermann Verlag, Berlin, 1989).V. V. Fedorcuk, Bicompacta with noncoinciding dimensionalities, Soviet Math. Doklady, 9/5 (1968), 1148–1150.K. P. Hart, J. Nagata and J.E. Vaughan (Eds.), Encyclopedia of General Topology (North-Holland, Amsterdam, 2004)

Selection principles and double sequences II

This paper is a continuation of the research on selection properties of certain classes of double sequences of positive real numbers that was began in [6].Publishe

Statistical Convergence in Function Spaces

We study statistical versions of several classical kinds of convergence of sequences of functions between metric spaces (Dini, Arzelà, and Alexandroff) in different function spaces. Also, we discuss a statistical approach to recently introduced notions of strong uniform convergence and exhaustiveness

Decomposition of topologies which characterize the upper and lower semicontinuous limits of functions. Abstract and Applied Analysis

We present a decomposition of two topologies which characterize the upper and lower semicontinuity of the limit function to visualize their hidden and opposite roles with respect to the upper and lower semicontinuity and consequently the continuity of the limit. We show that from the statistical point of view there is an asymmetric role of the upper and lower decomposition of the pointwise convergence with respect to the upper and lower decomposition of the sticking convergence and the semicontinuity of the limit. This role is completely hidden if we use the whole pointwise convergence. Moreover, thanks to this mirror effect played by these decompositions, the statistical pointwise convergence of a sequence of continuous functions to a continuous function in one of the two symmetric topologies, which are the decomposition of the sticking topology, automatically ensures the convergence in the whole sticking topology

Decomposition of Topologies Which Characterize the Upper and Lower Semicontinuous Limits of Functions

We present a decomposition of two topologies which characterize the upper and lower semicontinuity of the limit function to visualize their hidden and opposite roles with respect to the upper and lower semicontinuity and consequently the continuity of the limit. We show that (from the statistical point of view) there is an asymmetric role of the upper and lower decomposition of the pointwise convergence with respect to the upper and lower decomposition of the sticking convergence and the semicontinuity of the limit. This role is completely hidden if we use the whole pointwise convergence. Moreover, thanks to this mirror effect played by these decompositions, the statistical pointwise convergence of a sequence of continuous functions to a continuous function in one of the two symmetric topologies, which are the decomposition of the sticking topology, automatically ensures the convergence in the whole sticking topology

Michael spaces and Dowker planks

We investigate the Lindelöf property of Dowker planks. In particular, we give necessary conditions such that the product of a Dowker plank with the irrationals is not Lindelöf. We also show that if there exists a Michael space, then, under some conditions involving singular cardinals, there is one that is a Dowker plank

Michael spaces and Dowker planks

[EN] We investigate the Lindelöf property of Dowker planks. In particular, we give necessary conditions such that the product of a Dowker plank with the irrationals is not Lindelöf. We also show that if there exists a Michael space, then, under some conditions involving singular cardinals, there is one that is a Dowker plank.Caserta, A.; Watson, S. (2009). Michael spaces and Dowker planks. Applied General Topology. 10(2):245-267. doi:10.4995/agt.2009.1738.SWORD24526710