On Cartesian factors and the topological classification of linear metric spaces
On CE-images of the Hubert cube and characterization of Q-manifolds
On closed sets with convex projections in Hilbert space
Let k be a fixed natural number. We show that if C is a closed and nonconvex set in Hilbert space such that the closures of the projections onto all k-hyperplanes (planes with codimension k) are convex and proper, then C must contain a closed copy of Hilbert space. In order to prove this result we introduce for convex closed sets B the set consisting of all points of B that are extremal with respect to projections onto k-hyperplanes. We prove that is precisely the intersection of all k-imitations...
On Countable Dense and Strong Local Homogeneity
We present an example of a connected, Polish, countable dense homogeneous space X that is not strongly locally homogeneous. In fact, a nontrivial homeomorphism of X is the identity on no nonempty open subset of X.
On extension of functors
A. Chigogidze defined for each normal functor on the category Comp an extension which is a normal functor on the category Tych. We consider this extension for any functor on the category Comp and investigate which properties it preserves from the definition of normal functor. We investigate as well some topological properties of such extension.
On infinite-dimensional manifolds
On some applications of infinite-dimensional manifolds to the theory of shape
On the homotopical classification of DJ-mappings of infnitely dimensional spheres
On the stratification of the orbit space for the action of automorphisms on connections [Book]
On the structure of Hilbert cube manifolds
On universality of countable and weak products of sigma hereditarily disconnected spaces
Suppose a metrizable separable space Y is sigma hereditarily disconnected, i.e., it is a countable union of hereditarily disconnected subspaces. We prove that the countable power of any subspace X ⊂ Y is not universal for the class ₂ of absolute -sets; moreover, if Y is an absolute -set, then contains no closed topological copy of the Nagata space = W(I,ℙ); if Y is an absolute -set, then contains no closed copy of the Smirnov space σ = W(I,0). On the other hand, the countable power of...
On universality of finite powers of locally path-connected meager spaces
It is shown that for every integer n the (2n+1)th power of any locally path-connected metrizable space of the first Baire category is 𝓐₁[n]-universal, i.e., contains a closed topological copy of each at most n-dimensional metrizable σ-compact space. Also a one-dimensional σ-compact absolute retract X is found such that the power X^{n+1} is 𝓐₁[n]-universal for every n.
Open Subsets of LF-spaces
Let F = ind lim Fₙ be an infinite-dimensional LF-space with density dens F = τ ( ≥ ℵ ₀) such that some Fₙ is infinite-dimensional and dens Fₙ = τ. It is proved that every open subset of F is homeomorphic to the product of an ℓ₂(τ)-manifold and (hence the product of an open subset of ℓ₂(τ) and ). As a consequence, any two open sets in F are homeomorphic if they have the same homotopy type.
Orbit spaces of the hyperspace of a graph which are Hilbert cubes