On the dimension of ordered spaces.
On the dimension of remainders in extensions of product spaces
On the dimension of the space of ℝ-places of certain rational function fields
We prove that for every n ∈ ℕ the space M(K(x 1, …, x n) of ℝ-places of the field K(x 1, …, x n) of rational functions of n variables with coefficients in a totally Archimedean field K has the topological covering dimension dimM(K(x 1, …, x n)) ≤ n. For n = 2 the space M(K(x 1, x 2)) has covering and integral dimensions dimM(K(x 1, x 2)) = dimℤ M(K(x 1, x 2)) = 2 and the cohomological dimension dimG M(K(x 1, x 2)) = 1 for any Abelian 2-divisible coefficient group G.
On the fundamental dimension of the Cartesian product of compacta with the fundamental dimension 2
On the Hausdorff Dimension of Topological Subspaces
It is shown that every Polish space X with admits a compact subspace Y such that where and denote the topological and Hausdorff dimensions, respectively.
On the hyperspaces of hereditarily indecomposable continua
On the product of separable metric spaces.
On the Separation Dimension of
We prove that , where trt stands for the transfinite extension of Steinke’s separation dimension. This answers a question of Chatyrko and Hattori.
On the structure of the intersection of two middle third Cantor sets.
Motivated by the study of planar homoclinic bifurcations, in this paper we describe how the intersection of two middle third Cantor sets changes as the sets are translated across each other. The resulting description shows that the intersection is never empty; in fact, the intersection can be either finite or infinite in size. We show that when the intersection is finite then the number of points in the intersection will be either 2n or 3 · 2n. We also explore the Hausdorff dimension of the intersection...
On the thickness of topological spaces
On transfinite dimension
On transfinite inductive compactness degree
On (transfinite) small inductive dimension of products
In this paper we study the behavior of the (transfinite) small inductive dimension on finite products of topological spaces. In particular we essentially improve Toulmin’s estimation [T] of for Cartesian products.
On universal infinite-dimensional spaces
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.
On weakly infinite-dimensional subspuees
We will construct weakly infinite-dimensional (in the sense of Y. Smirnov) spaces X and Y such that Y contains X topologically and and . Consequently, the subspace theorem does not hold for the transfinite dimension dim for weakly infinite-dimensional spaces.
On weakly n-dimensional spaces
On zero-dimensionality of subgroups of locally compact groups
Improving the recent result of the author we show that is equivalent to for every subgroup of a Hausdorff locally compact group .
On Δ-spaces and fundamental dimension in the sense of Borsuk