A continuous surjection from the unit interval onto the unit square.
A continuous version of Liapunov's convexity theorem
A continuum without the fixed point property which is quasihomeomorphic with an AR-set
A continuum such that is not continuously homogeneous
A metric continuum is said to be continuously homogeneous provided that for every two points there exists a continuous surjective function such that . Answering a question by W.J. Charatonik and Z. Garncarek, in this paper we show a continuum such that the hyperspace of subcontinua of , , is not continuously homogeneous.
A contraction theorem in fuzzy metric spaces.
A contraction theorem in Menger probabilistic metric spaces.
A contribution to the descriptive theory of sets and spaces
A contribution to the theory of modular spaces
A contribution to the topological classification of the spaces Ср(X)
We prove that for each countably infinite, regular space X such that is a -space, the topology of is determined by the class of spaces embeddable onto closed subsets of . We show that , whenever Borel, is of an exact multiplicative class; it is homeomorphic to the absorbing set for the multiplicative Borel class if . For each ordinal α ≥ 2, we provide an example such that is homeomorphic to .
A contribution to topology in AST: Almost indiscernibilities
A contribution to topology in AST: Compactness
A convergence criterion for monotone global dynamical systems.
A convergence on Boolean algebras generalizing the convergence on the Aleksandrov cube
We compare the forcing-related properties of a complete Boolean algebra with the properties of the convergences (the algebraic convergence) and on generalizing the convergence on the Cantor and Aleksandrov cube, respectively. In particular, we show that is a topological convergence iff forcing by does not produce new reals and that is weakly topological if satisfies condition (implied by the -cc). On the other hand, if is a weakly topological convergence, then is a -cc algebra...
A converse of the Arsenin–Kunugui theorem on Borel sets with σ-compact sections
Let f be a Borel measurable mapping of a Luzin (i.e. absolute Borel metric) space L onto a metric space M such that f(F) is a Borel subset of M if F is closed in L. We show that then is a set for all except countably many y ∈ M, that M is also Luzin, and that the Borel classes of the sets f(F), F closed in L, are bounded by a fixed countable ordinal. This gives a converse of the classical theorem of Arsenin and Kunugui. As a particular case we get Taĭmanov’s theorem saying that the image of...
A Converse To A Generalized Banach Contraction Principle
A converse to a theorem of K. Kuratowski on parametrizations of compacta on the Cantor set
A correction to my paper:"Some topological extensions of plane geometry"
A Corson compact L-space from a Suslin tree
The completion of a Suslin tree is shown to be a consistent example of a Corson compact L-space when endowed with the coarse wedge topology. The example has the further properties of being zero-dimensional and monotonically normal.
A countable connected Urysohn space with a dispersion point that is regular almost everywhere
A countable dense homogeneous set of reals of size ℵ₁
We prove there is a countable dense homogeneous subspace of ℝ of size ℵ₁. The proof involves an absoluteness argument using an extension of the logic obtained by adding predicates for Borel sets.