We prove that every nonmetrizable compact connected Abelian group G has a family H of size |G|, the maximal size possible, consisting of proper dense pseudocompact subgroups of G such that H ∩ H'={0} for distinct H,H' ∈ H. An easy example shows that connectedness of G is essential in the above result. In the general case we establish that every nonmetrizable compact Abelian group G has a family H of size |G| consisting of proper dense pseudocompact subgroups of G such that each intersection H H'...

Arhangel’skii proved that if a first countable Hausdorff space is Lindelöf, then its cardinality is at most $2^{\aleph ₀}$. Such a clean upper bound for Lindelöf spaces in the larger class of spaces whose points are $G_{\delta }$ has been more elusive. In this paper we continue the agenda started by the second author, [Topology Appl. 63 (1995)], of considering the cardinality problem for spaces satisfying stronger versions of the Lindelöf property. Infinite games and selection principles, especially the Rothberger property,...

In [1] the author showed that if there is a cardinal κ such that $2^{\kappa }={\kappa }^{+}$ then there exists a completely regular space without dense 0-dimensional subspaces. This was a solution of a problem of Arkhangel’skiĭ. Recently Arkhangel’skiĭ asked the author whether one can generalize this result by constructing a completely regular space without dense totally disconnected subspaces, and whether such a space can have a structure of a linear space. The purpose of this paper is to show that indeed such a space can...

The cardinal functions of pseudocharacter, closed pseudocharacter, and character are used to examine H-closed spaces and to contrast the differences between H-closed and minimal Hausdorff spaces. An H-closed space $X$ is produced with the properties that $\left|X\right|>2^{2^{\psi \left(X\right)}}$ and $\overline{\psi }\left(X\right)>2^{\psi \left(X\right)}$.

We work towards establishing that if it is consistent that there is a supercompact cardinal then it is consistent that every locally compact perfectly normal space is paracompact. At a crucial step we use some still unpublished results announced by Todorcevic. Modulo this and the large cardinal, this answers a question of S. Watson. Modulo these same unpublished results, we also show that if it is consistent that there is a supercompact cardinal, it is consistent that every locally compact space...

We consider $M$-mappings which include continuous mappings of spaces onto topological groups and continuous mappings of topological groups elsewhere. It is proved that if a space $X$ is an image of a product of Lindelöf $\Sigma$-spaces under an $M$-mapping then every regular uncountable cardinal is a weak precaliber for $X$, and hence $X$ has the Souslin property. An image $X$ of a Lindelöf space under an $M$-mapping satisfies $ce{l}_{\omega }X\le 2^{\omega }$. Every $M$-mapping takes a $\Sigma \left({\aleph }_0\right)$-space to an ${\aleph }_0$-cellular space. In each of these results, the cellularity...

We study free sequences and related notions on Boolean algebras. A free sequence on a BA $A$ is a sequence $\langle a_{\xi }:\xi <\alpha \rangle$ of elements of $A$, with $\alpha$ an ordinal, such that for all $F,G\in {\left[\alpha \right]}^{<\omega }$ with $F<G$ we have ${\prod }_{\xi \in F}{a}_{\xi }·{\prod }_{\xi \in G}-a_{\xi }\ne 0$. A free sequence of length $\alpha$ exists iff the Stone space $Ult\left(A\right)$ has a free sequence of length $\alpha$ in the topological sense. A free sequence is maximal iff it cannot be extended at the end to a longer free sequence. The main notions studied here are the spectrum function $${𝔣}_{sp}\left(A\right)=\left\{\right|\alpha |:A\text{has}\text{an}\text{infinite}\text{maximal}\text{free}\text{sequence}\text{of}\text{length}\alpha \}$$ and the associated min-max function $$𝔣\left(A\right)=min\left({𝔣}_{sp}\left(A\right)\right).$$ Among the results...

In this paper we consider the point character of metric spaces. This parameter which is a uniform version of dimension, was introduced in the context of uniform spaces in the late seventies by Jan Pelant, Cardinal reflections and point-character of uniformities, Seminar Uniform Spaces (Prague, 1973–1974), Math. Inst. Czech. Acad. Sci., Prague, 1975, pp. 149–158. Here we prove for each cardinal $\kappa$, the existence of a metric space of cardinality and point character $\kappa$. Since the point character can...

It is proved that the product of two pseudo radial compact spaces is pseudo radial provided that one of them is monolithic.

We consider the question of when $X_M=X$, where $X_M$ is the elementary submodel topology on X ∩ M, especially in the case when $X_M$ is compact.

A new topological cardinal invariant is defined; it may be considered as a weaker form of the Lindelöf degree.

For Tychonoff $X$ and $\alpha$ an infinite cardinal, let $\alpha defX:=$ the minimum number of $\alpha$ cozero-sets of the Čech-Stone compactification which intersect to $X$ (generalizing $ℝ$-defect), and let $rtX:={min}_{\alpha }max(\alpha ,\alpha defX)$. Give $C\left(X\right)$ the compact-open topology. It is shown that $\tau C\left(X\right)\le n\chi C\left(X\right)\le rtX=max\left(L\right(X),L(X)defX)$, where: $\tau$ is tightness; $n\chi$ is the network character; $L\left(X\right)$ is the Lindel"of number. For example, it follows that, for $X$ Čech-complete, $\tau C\left(X\right)=L\left(X\right)$. The (apparently new) cardinal functions $n\chi C$ and $rt$ are compared with several others.

It is well-known that the concentric circle space has no $G_{\delta }$-diagonal nor any countable point-separating open cover. In this paper, we reveal two new properties of the concentric circle space, which are the weak versions of $G_{\delta }$-diagonal and countable point-separating open cover. Then we introduce two new cardinal functions and sharpen some known cardinal inequalities.

It is known that every remainder of a topological group is Lindelöf or pseudocompact. Motivated by this result, we study in this paper when a topological group $G$ has a normal remainder. In a previous paper we showed that under mild conditions on $G$, the Continuum Hypothesis implies that if the Čech-Stone remainder $G^{*}$ of $G$ is normal, then it is Lindelöf. Here we continue this line of investigation, mainly for the case of precompact groups. We show that no pseudocompact group, whose weight is uncountable...

We investigate the problem if every compact space K carrying a Radon measure of Maharam type κ can be continuously mapped onto the Tikhonov cube ${[0,1]}^{\kappa }$ (κ being an uncountable cardinal). We show that for κ ≥ cf(κ) ≥ κ this holds if and only if κ is a precaliber of measure algebras. Assuming that there is a family of ${\omega }_1$ null sets in $2^{\omega 1}$ such that every perfect set meets one of them, we construct a compact space showing that the answer to the above problem is “no” for κ = ω. We also give alternative proofs...