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...

The Katětov ordering of two maximal almost disjoint (MAD) families $𝒜$ and $ℬ$ is defined as follows: We say that $𝒜{\le }_Kℬ$ if there is a function $f:\omega \to \omega$ such that $f^{-1}\left(A\right)\in ℐ\left(ℬ\right)$ for every $A\in ℐ\left(𝒜\right)$. In [Garcia-Ferreira S., Hrušák M., Ordering MAD families a la Katětov, J. Symbolic Logic 68 (2003), 1337–1353] a MAD family is called $K$-uniform if for every $X\in ℐ{\left(𝒜\right)}^{+}$, we have that ${𝒜|}_X{\le }_K𝒜$. We prove that CH implies that for every $K$-uniform MAD family $𝒜$ there is a $P$-point $p$ of ${\omega }^{*}$ such that the set of all Rudin-Keisler predecessors of $p$ is dense in the...

A connected topological space $Z$ is unicoherent provided that if $Z=A\cup B$ where $A$ and $B$ are closed connected subsets of $Z$, then $A\cap B$ is connected. Let $Z$ be a unicoherent space, we say that $z\in Z$ makes a hole in $Z$ if $Z-\left\{z\right\}$ is not unicoherent. In this work the elements that make a hole to the cone and the suspension of a metric space are characterized. We apply this to give the classification of the elements of hyperspaces of some continua that make them hole.

In this paper $ss$-quotient maps and $ssq$-spaces are introduced. It is shown that (1) countable tightness is characterized by $ss$-quotient maps and quotient maps; (2) a space has countable tightness if and only if it is a countably bi-quotient image of a locally countable space, which gives an answer for a question posed by F. Siwiec in 1975; (3) $ssq$-spaces are characterized as the $ss$-quotient images of metric spaces; (4) assuming $2^{\omega }<2^{{\omega }_1}$, a compact $T_2$-space is an $ssq$-space if and only if every countably compact subset...

In this paper we improve some mapping theorems on $\aleph$-spaces. For instance we show that an $\aleph$-space is preserved by a closed and countably bi-quotient map. This is an improvement of Yun Ziqiu’s theorem: an $\aleph$-space is preserved by a closed and open map.