$\alpha$-normality and $\beta$-normality are properties generalizing normality of topological spaces. They consist in separating dense subsets of closed disjoint sets. We construct an example of a Tychonoff $\beta$-normal non-normal space and an example of a Hausdorff $\alpha$-normal non-regular space.

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...

Given a space $X$, its $G_{\delta }$-subsets form a basis of a new space $X_{\omega }$, called the $G_{\delta }$-modification of $X$. We study how the assumption that the $G_{\delta }$-modification $X_{\omega }$ is homogeneous influences properties of $X$. If $X$ is first countable, then $X_{\omega }$ is discrete and, hence, homogeneous. Thus, $X_{\omega }$ is much more often homogeneous than $X$ itself. We prove that if $X$ is a compact Hausdorff space of countable tightness such that the $G_{\delta }$-modification of $X$ is homogeneous, then the weight $w\left(X\right)$ of $X$ does not exceed $2^{\omega }$ (Theorem 1)....

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...