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.