The Borsuk homotopy extension theorem without the binormality condition
The box product of countably many metrizable spaces need not be normal
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 classes of mutual compactificability.
The common division topology on
A topological space is totally Brown if for each and every nonempty open subsets of we have . Totally Brown spaces are connected. In this paper we consider a topology on the set of natural numbers. We then present properties of the topological space , some of them involve the closure of a set with respect to this topology, while others describe subsets which are either totally Brown or totally separated. Our theorems generalize results proved by P. Szczuka in 2013, 2014, 2016 and by...
The compact open topology for semigroups of continuous selfmaps.
The compactificability classes of certain spaces.
The compactificability classes: The behavior at infinity.
The compactness number of a compact topological space I
The concentration-compactness principle in the calculus of variations. The locally compact case, part 1
The connectedness of degeneracy loci
The connectedness of symmetric degeneracy loci: odd ranks. Appendix to "The connectedness of degeneracy loci" by L. W. Tu
The controlled separable projection property for Banach spaces
Let X, Y be two Banach spaces. We say that Y is a quasi-quotient of X if there is a continuous operator R: X → Y such that its range, R(X), is dense in Y. Let X be a nonseparable Banach space, and let U, W be closed subspaces of X and Y, respectively. We prove that if X has the Controlled Separable Projection Property (CSPP) (this is a weaker notion than the WCG property) and Y is a quasi-quotient of X, then the structure of Y resembles the structure of a separable Banach space: (a) Y/W is norm-separable...
The covering property for σ-ideals of compact, sets
The covering property for σ-ideals of compact sets is an abstract version of the classical perfect set theorem for analytic sets. We will study its consequences using as a paradigm the σ-ideal of countable closed subsets of .
The dimension of remainders of rim-compact spaces
Answering a question of Isbell we show that there exists a rim-compact space X such that every compactification Y of X has dim(Y)≥ 1.
The (dis)connectedness of products of Hausdorff spaces in the box topology
In this paper the following two propositions are proved: (a) If , , is an infinite system of connected spaces such that infinitely many of them are nondegenerated completely Hausdorff topological spaces then the box product can be decomposed into continuum many disjoint nonempty open subsets, in particular, it is disconnected. (b) If , , is an infinite system of Brown Hausdorff topological spaces then the box product is also Brown Hausdorff, and hence, it is connected. A space is Brown if...
The dual group of a dense subgroup
Throughout this abstract, is a topological Abelian group and is the space of continuous homomorphisms from into the circle group in the compact-open topology. A dense subgroup of is said to determine if the (necessarily continuous) surjective isomorphism given by is a homeomorphism, and is determined if each dense subgroup of determines . The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is...
The Dual of a Space of Vector Measures.
The duality between lattice-ordered monoids and ordered topological spaces.
The Dugundji extension property can fail in ωµ -metrizable spaces
We show that there exist -metrizable spaces which do not have the Dugundji extension property ( with the countable box topology is such a space). This answers a question posed by the second author in 1972, and shows that certain results of van Douwen and Borges are false.