On determining sets for the class of somewhat continuous functions
On dimensionally restricted maps
Let f: X → Y be a closed n-dimensional surjective map of metrizable spaces. It is shown that if Y is a C-space, then: (1) the set of all maps g: X → ⁿ with dim(f △ g) = 0 is uniformly dense in C(X,ⁿ); (2) for every 0 ≤ k ≤ n-1 there exists an -subset of X such that and the restriction is (n-k-1)-dimensional. These are extensions of theorems by Pasynkov and Toruńczyk, respectively, obtained for finite-dimensional spaces. A generalization of a result due to Dranishnikov and Uspenskij about...
On dyadic spaces and almost Milyutin spaces
On Eberlein compactifications of metrizable spaces
We prove that, for every finite-dimensional metrizable space, there exists a compactification that is Eberlein compact and preserves both the covering dimension and weight.
On elementary maps
On embeddability of cones in euclidean spaces
On embedding curves in two-dimensional polyhedra
On embedding of curves into two-dimensional polyhedra
On embeddings into where is Lindelöf
A.V. Arkhangel’skii asked that, is it true that every space of countable tightness is homeomorphic to a subspace (to a closed subspace) of where is Lindelöf? denotes the space of all continuous real-valued functions on a space with the topology of pointwise convergence. In this note we show that the two arrows space is a counterexample for the problem by showing that every separable compact linearly ordered topological space is second countable if it is homeomorphic to a subspace of ...
On entropy of patterns given by interval maps
Defining the complexity of a green pattern exhibited by an interval map, we give the best bounds of the topological entropy of a pattern with a given complexity. Moreover, we show that the topological entropy attains its strict minimum on the set of patterns with fixed eccentricity m/n at a unimodal X-minimal case. Using a different method, the last result was independently proved in[11].
On equalizers in the category of frames with weakly open homomorphisms
On essential ring embeddings and the epimorphic hull of .
On evolution of small spheres in the phase space of a dynamical system*
We study the connection between the entropy of a dynamical system and the boundary distortion rate of regions in the phase space of the system.
On expansions of a Urysohn space to a regular space.
On extending functions from an open set to with applications
For and open in , let be the ring of real valued functions on with the first derivatives continuous. It is shown that for there is with and with . The function and its derivatives are not assumed to be bounded on . The function is constructed using splines based on the Mollifier function. Some consequences about the ring are deduced from this, in particular that .
On extending homeomorphisms on zero-dimensional spaces
On extending transitive homeomorphisms from the Cantor set to the product of two Cantor sets
On Extensions of Multivalued Mappings of Topological Spaces
On extensions of uniformly continuous Banach-space-valued mappings
On fine shape theory