We construct a consistent example of a normal locally compact metacompact space which is not paracompact, answering a question of A. V. Arkhangel’skiĭ and F. Tall. An interplay between a tower in P(ω)/Fin, an almost disjoint family in ${\left[\omega \right]}^{\omega }$, and a version of an (ω,1)-morass forms the core of the proof. A part of the poset which forces the counterexample can be considered a modification of a poset due to Judah and Shelah for obtaining a Q-set by a countable support iteration.

We show that MA${}_{\sigma -centered}\left({\omega }_1\right)$ implies that normal locally compact metacompact spaces are paracompact, and that MA(${\omega }_1$) implies normal locally compact metalindelöf spaces are paracompact. The latter result answers a question of S. Watson. The first result implies that there is a model of set theory in which all normal locally compact metacompact spaces are paracompact, yet there is a normal locally compact metalindelöf space which is not paracompact.

We establish a relation between covering properties (e.gĿindelöf degree) of two standard topological spaces (Lemmas 4 and 5). Some cardinal inequalities follow as easy corollaries.

According to a result of Kočinac and Scheepers, the Hurewicz covering property is equivalent to a somewhat simpler selection property: For each sequence of large open covers of the space one can choose finitely many elements from each cover to obtain a groupable cover of the space. We simplify the characterization further by omitting the need to consider sequences of covers: A set of reals X has the Hurewicz property if, and only if, each large open cover of X contains a groupable subcover. This...

2000 Mathematics Subject Classification: 54C35, 54D20, 54C60.Two Tychonoff spaces X and Y are said to be l-equivalent (u-equivalent) if Cp(X) and Cp(Y) are linearly (uniformly) homeomorphic. N. V. Velichko proved that countable Lindelöf number is preserved by the relation of l-equivalence. A. Bouziad strengthened this result and proved that any Lindelöf number is preserved by the relation of l-equivalence. In this paper it has been proved that the Lindelöf number greater than continuum is preserved...

We introduce and study, following Z. Frol’ık, the class $ℬ\left(𝒫\right)$ of regular $P$-spaces $X$ such that the product $X\times Y$ is pseudo-${\aleph }_1$-compact, for every regular pseudo-${\aleph }_1$-compact $P$-space $Y$. We show that every pseudo-${\aleph }_1$-compact space which is locally $ℬ\left(𝒫\right)$ is in $ℬ\left(𝒫\right)$ and that every regular Lindelöf $P$-space belongs to $ℬ\left(𝒫\right)$. It is also proved that all pseudo-${\aleph }_1$-compact $P$-groups are in $ℬ\left(𝒫\right)$. The problem of characterization of subgroups of $ℝ$-factorizable (equivalently, pseudo-${\aleph }_1$-compact) $P$-groups is considered as well. We give some necessary...

A topological space (T,τ) is said to be fragmented by a metric d on T if each non-empty subset of T has non-empty relatively open subsets of arbitrarily small d-diameter. The basic theorem of the present paper is the following. Let (M,ϱ) be a metric space with ϱ bounded and let D be an arbitrary index set. Then for a compact subset K of the product space $M^D$ the following four conditions are equivalent: (i) K is fragmented by $d_D$, where, for each S ⊂ D, $d_S(x,y)=sup\varrho \left(x\right(t),y(t\left)\right):t\in S$. (ii) For each countable subset A of D, $(K,d_A)$ is...

We prove that the one-point Lindelöfication of a discrete space of cardinality ω 1 is homeomorphic to a subspace of C p (X) for some hereditarily Lindelöf space X if the axiom [...] holds.

In Dually discrete spaces, Topology Appl. 155 (2008), 1420–1425, Alas et. al. proved that ordinals are hereditarily dually discrete and asked whether the product of two ordinals has the same property. In Products of certain dually discrete spaces, Topology Appl. 156 (2009), 2832–2837, Peng proved a number of partial results and left open the question of whether the product of two stationary subsets of ${\omega }_1$ is dually discrete. We answer the first question affirmatively and as a consequence also give...

Several topological properties lying between the submetrizability and the $G_{\delta }$-diagonal property are studied. We are mostly interested in their relationship to each other and to the submetrizability. The first example of a Tychonoff space with a regular $G_{\delta }$-diagonal but without a zero-set diagonal is given. The same example shows that a Tychonoff separable space with a regular $G_{\delta }$-diagonal need not be submetrizable. We give a necessary and sufficient condition for submetrizability of a regular separable...