On the characterization of weak closure in Hilbert space
On the Clarke’s generalized jacobian
On the class of all spaces of weight not greater than ω_1 whose Cartesian product with every Lindelöf space is Lindelöf
On the class of continuous images of Valdivia compacta.
On the class of functions having infinite limit on a given set
On the classification of inverse limits of tent maps
Let and be tent maps on the unit interval. In this paper we give a new proof of the fact that if the critical points of and are periodic and the inverse limit spaces and are homeomorphic, then s = t. This theorem was first proved by Kailhofer. The new proof in this paper simplifies the proof of Kailhofer. Using the techniques of the paper we are also able to identify certain isotopies between homeomorphisms on the inverse limit space.
On the classification of locally compact separable metric spaces
On the cliquish, quasicontinuous and measurable selections
The purpose of this paper is the investigation of the necessary and sufficient conditions under which a given multifunctions admits a cliquish and measurable selection. Our investigation also covers the search for quasicontinuous selections for multifunctions which are continuous with respect to the generalized notion of the semi-quasicontinuity.
On the closure of Baire classes under transfinite convergences
Let X be a Polish space and Y be a separable metric space. For a fixed ξ < ω₁, consider a family of Baire-ξ functions. Answering a question of Tomasz Natkaniec, we show that if for a function f: X → Y, the set is finite for every x ∈ X, then f itself is necessarily Baire-ξ. The proof is based on a characterization of sets which can be interesting in its own right.
On the closure of the bicyclic semigroup.
On the co-completeness of the category of Hausdorff uniform spaces.
On the combinatorial principle P(c)
On the common fixed point for commuting Lipschitz functions
On the compactification of closure algebras
On the Compactness and Countable Compactness of in ZF
In the framework of ZF (Zermelo-Fraenkel set theory without the Axiom of Choice) we provide topological and Boolean-algebraic characterizations of the statements " is countably compact" and " is compact"
On the completeness of a topological group of functions.
On the completeness of localic groups
The main purpose of this paper is to show that any localic group is complete in its two-sided uniformity, settling a problem open since work began in this area a decade ago. In addition, a number of other results are established, providing in particular a new functor from topological to localic groups and an alternative characterization of -groups.
On the complexity of Hamel bases of infinite-dimensional Banach spaces
We call a subset S of a topological vector space V linearly Borel if for every finite number n, the set of all linear combinations of S of length n is a Borel subset of V. It is shown that a Hamel basis of an infinite-dimensional Banach space can never be linearly Borel. This answers a question of Anatoliĭ Plichko.
On the complexity of some -ideals of -P-porous sets
Let be a porosity-like relation on a separable locally compact metric space . We show that the -ideal of compact --porous subsets of (under some general conditions on and ) forms a -complete set in the hyperspace of all compact subsets of , in particular it is coanalytic and non-Borel. Our general results are applicable to most interesting types of porosity. It is shown in the cases of the -ideals of -porous sets, --porous sets, -strongly porous sets, -symmetrically porous sets...
On the complexity of subspaces of
Let (X,τ) be a countable topological space. We say that τ is an analytic (resp. Borel) topology if τ as a subset of the Cantor set (via characteristic functions) is an analytic (resp. Borel) set. For example, the topology of the Arkhangel’skiĭ-Franklin space is . In this paper we study the complexity, in the sense of the Borel hierarchy, of subspaces of . We show that has subspaces with topologies of arbitrarily high Borel rank and it also has subspaces with a non-Borel topology. Moreover,...