Tensor products with bounded continuous functions.
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 ANR-property of hyperspaces with the Attouch-Wets topology
We characterize metric spaces whose hyperspaces of non-empty closed, bounded, compact or finite subsets, endowed with the Attouch-Wets topology, are absolute (neighborhood) retracts.
The AR-Property of the spaces of closed convex sets
Let , and be the spaces of all non-empty closed convex sets in a normed linear space X admitting the Hausdorff metric topology, the Attouch-Wets topology and the Wijsman topology, respectively. We show that every component of and the space are AR. In case X is separable, is locally path-connected.
The axiom of choice and two particular forms of Tychonoff theorem
The Baire property in remainders of topological groups and other results
It is established that a remainder of a non-locally compact topological group has the Baire property if and only if the space is not Čech-complete. We also show that if is a non-locally compact topological group of countable tightness, then either is submetrizable, or is the Čech-Stone remainder of an arbitrary remainder of . It follows that if and are non-submetrizable topological groups of countable tightness such that some remainders of and are homeomorphic, then the spaces...
The Borel structure of the collections of sub-self-similar sets and super-self-similar sets.
The box product of countably many metrizable spaces need not be normal
The cartesian closed hull of the category of approach spaces
The cartesian closed topological hull of the category of completely regular filterspaces
The category of compactifications and its coreflections
We define “the category of compactifications”, which is denoted CM, and consider its family of coreflections, denoted corCM. We show that corCM is a complete lattice with bottom the identity and top an interpretation of the Čech–Stone . A corCM implies the assignment to each locally compact, noncompact a compactification minimum for membership in the “object-range” of . We describe the minimum proper compactifications of locally compact, noncompact spaces, show that these generate the atoms...
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 commuting of coreflectors in uniform spaces with completion
The completion monad and its algebra
The convergence approach to exponentiable maps.
The convexity of the subset space of a metric space
The covering dimension of product of generalized Sorgenfrey lines
The dimension of hyperspaces of non-metrizable continua
We prove that, for any Hausdorff continuum X, if dim X ≥ 2 then the hyperspace C(X) of subcontinua of X is not a C-space; if dim X = 1 and X is hereditarily indecomposable then either dim C(X) = 2 or C(X) is not a C-space. This generalizes some results known for metric continua.
The Dimension of the Set of All Finite Subsets of the Continuum