Let $\Psi \left(\Sigma \right)$ be the Isbell-Mr’owka space associated to the $MAD$-family $\Sigma$. We show that if $G$ is a countable subgroup of the group $𝐒\left(\omega \right)$ of all permutations of $\omega$, then there is a $MAD$-family $\Sigma$ such that every $f\in G$ can be extended to an autohomeomorphism of $\Psi \left(\Sigma \right)$. For a $MAD$-family $\Sigma$, we set $Inv\left(\Sigma \right)=\{f\in 𝐒(\omega ):f[A]\in \Sigma$ for all $A\in \Sigma \}$. It is shown that for every $f\in 𝐒\left(\omega \right)$ there is a $MAD$-family $\Sigma$ such that $f\in Inv\left(\Sigma \right)$. As a consequence of this result we have that there is a $MAD$-family $\Sigma$ such that $n+A\in \Sigma$ whenever $A\in \Sigma$ and $n<\omega$, where $n+A=\{n+a:a\in A\}$ for $n<\omega$. We also notice that there is no $MAD$-family...

Let $X$ and $Y$ be vector spaces over the same field $K$. Following the terminology of Richard Arens [Pacific J. Math. 11 (1961), 9–23], a relation $F$ of $X$ into $Y$ is called linear if $\lambda F\left(x\right)\subset F\left(\lambda x\right)$ and $F\left(x\right)+F\left(y\right)\subset F(x+y)$ for all $\lambda \in K\setminus \left\{0\right\}$ and $x,y\in X$. After improving and supplementing some former results on linear relations, we show that a relation $\Phi$ of a linearly independent subset $E$ of $X$ into $Y$ can be extended to a linear relation $F$ of $X$ into $Y$ if and only if there exists a linear subspace $Z$ of $Y$ such that $\Phi \left(e\right)\in Y|Z$ for all $e\in E$. Moreover, if...

We consider when one-to-one continuous mappings can improve normality-type and compactness-type properties of topological spaces. In particular, for any Tychonoff non-pseudocompact space $X$ there is a $\mu$ such that $X^{\mu }$ can be condensed onto a normal ($\sigma$-compact) space if and only if there is no measurable cardinal. For any Tychonoff space $X$ and any cardinal $\nu$ there is a Tychonoff space $M$ which preserves many properties of $X$ and such that any one-to-one continuous image of $M^{\mu }$, $\mu \le \nu$, contains a...