We suggest a method of constructing decompositions of a topological space X having an open subset homeomorphic to the space (R-n , tau), where n is an integer greater than= 1 and tau is any admissible extension of the Euclidean topology of R-n (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.
We study the algebra of semigroups of sets (i.e. families of sets closed under finite unions) and its applications. For each n greater than 1 we produce two finite nested families of pairwise different semigroups of sets consisting of subsets of R" without the Baire property.
We obtain estimates of the small and large inductive dimensions ind and Ind of the union of two sets, outside the class of completely normal spaces. We show that, in the sense of the inductive dimensions ind0 and Ind0 introduced independently by Charalambous and Filippov, a compact completely normal space which is the union of two dense zero-dimensional subspaces can be infinite-dimensional. © 2004 Elsevier B.V. All rights reserved.
In this paper we improve two theorems for the small inductive dimension ind in the regular T-1-spaces: an addition theorem from [M.G Charalambous, V.A. Chatyrko. Some estimates of the inductive dimensions of the union of two sets, Topology Appl. 146/147 (2005) 227-238] and a product theorem from [V.A. Chatyrko. K.L. Kozlov. On (transfinite) small inductive dimension of products. Comment. Math. Univ. Carolin. 41 (3) (2000) 597-603].
In this paper we describe those locally compact noncompact separable metrizable spaces X for which the class R(X) of all metrizable remainders of X consists of all metrizable non-empty compacta. Then we show that for any pair X and X of locally compact noncompact connected separable metrizable spaces, either R(X) subset of R(X) or R(X) subset of R(X).
In this paper we prove that each subspace of an Alexandroff T-0-space is semi-T-1/2. In particular, any subspace of the folder X-n, where n is a positive integer and X is either the Khalimsky line (Z,tau(K)), the Marcus-Wyse plane (Z(2), tau(MW)) or any partially ordered set with the upper topology is semi-T-1/2. Then we study the basic properties of spaces possessing the axiom semi-T-1/2 such as finite productiveness and monotonicity.
A topological space is reversible if every continuous bijection f:X→X is a homeomorphism. There are many examples of reversiblespaces; in particular, Hausdorff compact spaces and locally Euclidean spaces are such. Chatyrko and Hattori observed, in a manuscript, that any product of topological spaces is non-reversible whenever at least one of the spaces is non-reversible and asked whether the topological product of two connected reversible spaces is reversible. The authors prove here that there are connected reversible spaces such that their product is not reversible. In fact, they construct a reversible space X which is a connected 2-manifold in R3 without boundary such that X×[0,1] is not reversible.
For each pair of positive integers k and m with k ≤ m there exists a separable metrizable space X (k, m) such that cmp X (k, m) = k and def X (k, m) = m. This solves Problem 6 from [J.M. Aarts, T. Nishiura, Dimension and Extensions, North-Holland, Amsterdam, 1993, p. 71]. © 2004 Elsevier B.V. All rights reserved.
On the set $\mathbb R$ of real numbers we consider a poset $\mathcal P_\tau(\mathbb R)$ (by inclusion) of topologies $\tau(A)$, where $A\subseteq \mathbb R$, such that $A_1\supseteq A_2$ iff $\tau(A_1)\subseteq \tau(A_2)$. The poset has the minimal element $\tau (\mathbb R)$, the Euclidean topology, and the maximal element $\tau (\emptyset)$, the Sorgenfrey topology. We are interested when two topologies $\tau_1$ and $\tau_2$ (especially, for $\tau_2 = \tau(\emptyset)$) from the poset define homeomorphic spaces $(\mathbb R, \tau_1)$ and $(\mathbb R, \tau_2)$. In particular, we prove that for a closed subset $A$ of $\mathbb R$ the space $(\mathbb R, \tau(A))$ is homeomorphic to the Sorgenfrey line $(\mathbb R, \tau(\emptyset))$ iff $A$ is countable. We study also common properties of the spaces $(\mathbb R, \tau(A)), A\subseteq \mathbb R$.
We present a unified approach to define dimension functions like trind, trindp, trt and p. We show how some similar facts on these functions can be proved similarly. Moreover, several new classes of infinite-dimensional spaces close to the classes of countable-dimensional and σ-hereditarily disconnected ones are introduced. We prove a compactification theorem for these classes.
We suggest two independent sufficient conditions on topological connected spaces with axioms lower than $T_3$, which imply disconnectedness, and one sufficient condition, which implies connectedness, of products of spaces endowed with the box topology.
In this article, we shall discuss the possibility of different presentations of (locally compact) spaces as unions or partitions of locally compact subspaces. © 2006 University of Houston.
We consider the following classical problems: (1) For what spaces X and Y the existence of continuous bijections of X onto Y and Y onto X implies or does not imply that the spaces are homeomorphic? (2) For what spaces X is each continuous bijection of X onto itself a homeomorphism? Some answers to the questions are suggested. (C) 2015 Elsevier B.V. All rights reserved.
We discuss an approach to a problem posed by A.V. Arhangelskii and E.K. van Douwen on a possibility to present a compact space as a continuous image of a homogeneous compact space. Then we suggest some ways of proving nonhomogeneity of tau-powers of a space X using points of local connectedness (or local contractibility) and components of path connectedness of X.