Sums of almost zero-dimensional spaces (original) (raw)

Some classes of topological spaces related to zero-sets

Applied General Topology, 2022

An almost P-space is a topological space in which every zero-set is regular-closed. We introduce a large class of spaces, C-almost P-space (briefly CAP-space), consisting of those spaces in which the closure of the interior of every zero-set is a zero-set. In this paper we study CAP-spaces. It is proved that if X is a dense and Z#-embedded subspace of a space T, then T is CAP if and only if X is a CAP and CRZ-extended in T (i.e, for each regular-closed zero-set Z in X, clTZ is a zero-set in T). In 6P.5 of [8] it was shown that a closed countable union of zero-sets need not be a zero-set. We call X a CZ-space whenever the closure of any countable union of zero-sets is a zero-set. This class of spaces contains the class of P-spaces, perfectly normal spaces, and is contained in the cozero complemented spaces and CAP-spaces. In this paper we study topological properties of CZ (resp. cozero complemented)-space and other classes of topological spaces near to them. Some algebraic and topol...

On the dimension of almost 𝑛-dimensional spaces

Proceedings of the American Mathematical Society, 1999

Oversteegen and Tymchatyn proved that homeomorphism groups of positive dimensional Menger compacta are 1 1 -dimensional by proving that almost 0 0 -dimensional spaces are at most 1 1 -dimensional. These homeomorphism groups are almost 0 0 -dimensional and at least 1 1 -dimensional by classical results of Brechner and Bestvina. In this note we prove that almost n n -dimensional spaces for n ≥ 1 n \geq 1 are n n -dimensional. As a corollary we answer in the affirmative an old question of R. Duda by proving that every hereditarily locally connected, non-degenerate, separable, metric space is 1 1 -dimensional.

On a weak sum theorem in dimension theory

Israel Journal of Mathematics, 1974

The above space (proved to be. in some sense, most simplel shows also that the dimension ind of a metric space can be raised by adjoining of a single point, a fact proved recently by E.K. Van Douwen and by T. Przymusifiski. Some maximality property of the family {X;Ind X = 0} is proved and conditions implying P-ind = P-lnd are given. "'*This is part of a research thesis at the Technion. Israel Institute of Technology. towards an M.Sc. degree, directed by Professor M. Reichaw.

Lacunary Almost Summability in Certain Linear Topological Spaces

2004

In this paper, the concept of lacunary almost summability of sequences in locally convex spaces has been defined and investigated. It is also proved Kojima-Schur and Silvarman- Toeplitz type theorems for lacunary almost conservatively and lacunary almost regularity of the which transform sequences in a Frechet space into a sequence in an other Frechet space. We also stated that the

Metrizable Subspaces of Spaces Having

2015

Abstract. In [1], van Douwen, Lutzer, Pelant, and Reed asked if every regular space with a point-countable base can written as the union of c-many metrizable subspaces. They also asked the same question for closed metrizable subspaces. In this note, we construct a counterexample to the second question; the rst question remains open.

On spaces without non-trivial subcontinua and the dimension of their products

Topology and its Applications, 2004

We introduce splintered and strongly splintered spaces. They are generalizations of both almost zero-dimensional spaces and weakly 1-dimensional spaces. We prove that there are n-dimensional strongly splintered spaces for every n, and that there is a 1-dimensional splintered space X such that dim X n = n for every n. This solves a problem in the literature. Finally, we correct a flaw in an argument of Tomaszewski in his product formula for the dimension of the product of a weakly n-dimensional and a weakly m-dimensional space.