Advanced Search

The notion of the ordinal product of a transfinite sequence of topological spaces which is an extension of the finite product operation is introduced. The dimensions of finite and infinite ordinal products are estimated. In particular, the dimensions of ordinary products of Smirnov's [S] and Henderson's [He1] compacta are calculated.

We discuss the exactness of estimates in the finite sum theorems for transfinite inductive dimensions trind and trInd. The technique obtained gives an opportunity to repeat and sometimes strengthen some well known results about compacta with trind ≠ trInd. In particular we improve an estimate of the small transfinite inductive dimension of Smirnov’s compacta $S^{\alpha },\alpha <{\omega }_1$, given by Luxemburg [Lu2].

For each ordinal 1 ≤ α < ω₁ we present separable metrizable spaces $X_{\alpha }$, $Y_{\alpha }$ and $Z_{\alpha }$ such that (i) $f{X}_{\alpha },f{Y}_{\alpha },f{Z}_{\alpha }=\omega ₀$, where f is either trdef or ₀-trsur, (ii) $A\left(\alpha \right)-trind{X}_{\alpha }=\infty$ and $M\left(\alpha \right)-trind{X}_{\alpha }=-1$, (iii) $A\left(\alpha \right)-trind{Y}_{\alpha }=-1$ and $M\left(\alpha \right)-trind{Y}_{\alpha }=\infty$, and (iv) $A\left(\alpha \right)-trind{Z}_{\alpha }=M\left(\alpha \right)-trind{Z}_{\alpha }=\infty$ and $A(\alpha +1)\cap M(\alpha +1)-trind{Z}_{\alpha }=-1$. We also show that there exists no separable metrizable space $W_{\alpha }$ with $A\left(\alpha \right)-trind{W}_{\alpha }\ne \infty$, $M\left(\alpha \right)-trind{W}_{\alpha }\ne \infty$ and $A\left(\alpha \right)\cap M\left(\alpha \right)-trind{W}_{\alpha }=\infty$, where A(α) (resp. M(α)) is the absolutely additive (resp. multiplicative) Borel class.

In this paper the following two propositions are proved: (a) If $X_{\alpha }$, $\alpha \in A$, is an infinite system of connected spaces such that infinitely many of them are nondegenerated completely Hausdorff topological spaces then the box product ${\square }_{\alpha \in A}{X}_{\alpha }$ can be decomposed into continuum many disjoint nonempty open subsets, in particular, it is disconnected. (b) If $X_{\alpha }$, $\alpha \in A$, is an infinite system of Brown Hausdorff topological spaces then the box product ${\square }_{\alpha \in A}{X}_{\alpha }$ is also Brown Hausdorff, and hence, it is connected. A space is Brown if...

The notion of locally $r$-incomparable families of compacta was introduced by K. Borsuk [KB]. In this paper we shall construct uncountable locally $r$-incomparable families of different types of finite-dimensional Cantor manifolds.

On the set $ℝ$ of real numbers we consider a poset ${𝒫}_{\tau }\left(ℝ\right)$ (by inclusion) of topologies $\tau \left(A\right)$, where $A\subseteq ℝ$, such that $A_1\supseteq A_2$ iff $\tau \left(A_1\right)\subseteq \tau \left(A_2\right)$. The poset has the minimal element $\tau \left(ℝ\right)$, the Euclidean topology, and the maximal element $\tau \left(\varnothing \right)$, the Sorgenfrey topology. We are interested when two topologies ${\tau }_1$ and ${\tau }_2$ (especially, for ${\tau }_2=\tau \left(\varnothing \right)$) from the poset define homeomorphic spaces $(ℝ,{\tau }_1)$ and $(ℝ,{\tau }_2)$. In particular, we prove that for a closed subset $A$ of $ℝ$ the space $(ℝ,\tau (A\left)\right)$ is homeomorphic to the Sorgenfrey line $(ℝ,\tau (\varnothing \left)\right)$ iff $A$ is countable. We study also common properties...

Let $Zₙ={[0,1]}^{n+1}\setminus (0,1)ⁿ\times 0$. Then cmp Zₙ < def Zₙ for n ≥ 5. This is the answer to a question posed by de Groot and Nishiura [GN] for n ≥ 5.

In this paper we study the behavior of the (transfinite) small inductive dimension $\left(trind\right)$ $ind$ on finite products of topological spaces. In particular we essentially improve Toulmin’s estimation [T] of $trind$ for Cartesian products.

We suggest a method of constructing decompositions of a topological space X having an open subset homeomorphic to the space (ℝⁿ,τ), where n is an integer ≥ 1 and τ is any admissible extension of the Euclidean topology of ℝⁿ (in particular, X can be a finite-dimensional separable metrizable manifold), into a countable family ℱ of sets (dense in X and zero-dimensional in the case of manifolds) such that the union of each non-empty proper subfamily of ℱ does not have the Baire property in X.