In this paper, the concepts of $m^{*}r$-fuzzy $\tilde{g}$-open $F_{\sigma }$ sets and $m^{*}$-fuzzy basically disconnected spaces are introduced in the sense of Šostak and Ramadan. Some interesting properties and characterizations are studied. Tietze extension theorem for $m^{*}$-fuzzy basically disconnected spaces is discussed.

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 prove that, assuming MA, every crowded $T_0$ space $X$ is $\omega$-resolvable if it satisfies one of the following properties: (1) it contains a $\pi$-network of cardinality $<𝔠$ constituted by infinite sets, (2) $\chi \left(X\right)<𝔠$, (3) $X$ is a $T_2$ Baire space and $c\left(X\right)\le {\aleph }_0$ and (4) $X$ is a $T_1$ Baire space and has a network $𝒩$ with cardinality $<𝔠$ and such that the collection of the finite elements in it constitutes a $\sigma$-locally finite family. Furthermore, we prove that the existence of a $T_1$ Baire irresolvable space is equivalent to the existence of...

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...

Maximal pseudocompact spaces (i.e. pseudocompact spaces possessing no strictly stronger pseudocompact topology) are characterized. It is shown that submaximal pseudocompact spaces whose pseudocompact subspaces are closed need not be maximal pseudocompact. Various techniques for constructing maximal pseudocompact spaces are described. Maximal pseudocompactness is compared to maximal feeble compactness.

We show that the existence of a non-trivial category base on a set of regular cardinality with each subset being Baire is equiconsistent to the existence of a measurable cardinal.

We investigate whether in the setting of approach spaces there exist measures of relative compactness, (relative) sequential compactness and (relative) countable compactness in the same vein as Kuratowski's measure of compactness. The answer is yes. Not only can we prove that such measures exist, but we can give usable formulas for them and we can prove that they behave nicely with respect to each other in the same way as the classical notions.

We construct two examples of a compact, 0-dimensional space which supports a Radon probability measure whose measure algebra is isomorphic to the measure algebra of $2^{{\omega }_1}$. The first construction uses ♢ to produce an S-space with no convergent sequences in which every perfect set is a $G_{\delta }$. A space with these properties must be both hereditarily normal and hereditarily countably paracompact. The second space is constructed under CH and is both HS and HL.

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...

A locallic version of Hager’s metric-fine spaces is presented. A general definition of $𝒜$-fineness is given and various special cases are considered, notably $𝒜=$ all metric frames, $𝒜=$ complete metric frames. Their interactions with each other, quotients, separability, completion and other topological properties are discussed.