Symmetry conditions on the coincidence of some notions of quasi-uniform completeness.
Szpilrajn type theorem for concentration dimension
Let X be a locally compact, separable metric space. We prove that , where and stand for the concentration dimension and the topological dimension of X, respectively.
and objects at in the category of proximity spaces
In previous papers, various notions of pre-Hausdorff, Hausdorff and regular objects at a point in a topological category were introduced and compared. The main objective of this paper is to characterize each of these notions of pre-Hausdorff, Hausdorff and regular objects locally in the category of proximity spaces. Furthermore, the relationships that arise among the various , , , structures at a point are investigated. Finally, we examine the relationships between the generalized separation...
- pairwise continuous functions and bitopological separation axioms
Tangency relations for sets in some classes in generalized metric spaces
The algebraic dimension of linear metric spaces and Baire properties of their hyperspaces.
Answering a question of Halbeisen we prove (by two different methods) that the algebraic dimension of each infinite-dimensional complete linear metric space X equals the size of X. A topological method gives a bit more: the algebraic dimension of a linear metric space X equals |X| provided the hyperspace K(X) of compact subsets of X is a Baire space. Studying the interplay between Baire properties of a linear metric space X and its hyperspace, we construct a hereditarily Baire linear metric space...
The approximation and extension of uniformly continuous Banach space valued mappings
The Baire Set Fixing Property and Uniform Continuity in Topological Groups.
The Banach–Mazur game and σ-porosity
It is well known that the sets of the first category in a metric space can be described using the so-called Banach-Mazur game. We will show that if we change the rules of the Banach-Mazur game (by forcing the second player to choose large balls) then we can describe sets which can be covered by countably many closed uniformly porous sets. A characterization of σ-very porous sets and a sufficient condition for σ-porosity are also given in the terminology of games.
The box product of countably many metrizable spaces need not be normal
The cartesian closed hull of the category of approach spaces
The category of uniform frames
The category of uniform spaces as a completion of the category of metric spaces
A criterion for the existence of an initial completion of a concrete category universal w.r.tḟinite products and subobjects is presented. For metric spaces and uniformly continuous maps this completion is the category of uniform spaces.
The class of -spaces is invariant of closed mappings with Lindelöf fibres
The commuting of coreflectors in uniform spaces with completion
The completion monad and its algebra
The complexity of -discretely decomposable families in uniform spaces
The Conley index for flows preserving generalized symmetries
Topological spaces with generalized symmetries are defined and extensions of the Conley index of a compact isolated invariant set of the flow preserving the structures introduced are proposed. One of the two new indexes is constructed with no additional assumption on the examined set in terms of symmetry invariance.
The convergence space of minimal usco mappings
A convergence structure generalizing the order convergence structure on the set of Hausdorff continuous interval functions is defined on the set of minimal usco maps. The properties of the obtained convergence space are investigated and essential links with the pointwise convergence and the order convergence are revealed. The convergence structure can be extended to a uniform convergence structure so that the convergence space is complete. The important issue of the denseness of the subset of all...
The convexity of the subset space of a metric space