We introduce some practical calculation of the Riesz angles in Orlicz sequence spaces equipped with Luxemburg norm and Orlicz norm. For an $N$-function $\Phi \left(u\right)$ whose index function is monotonous, the exact value $a\left(l^{\left(\Phi \right)}\right)$ of the Orlicz sequence space with Luxemburg norm is $a\left(l^{\left(\Phi \right)}\right)=2^{\frac{1}{C_{\Phi }^0}}$ or $a\left(l^{\left(\Phi \right)}\right)=\frac{{\Phi }^{-1}\left(1\right)}{{\Phi }^{-1}\left(\frac{1}{2}\right)}$. The Riesz angles of Orlicz space $l^{\Phi }$ with Orlicz norm has the estimation $max(2{\beta }_{\Psi }^0,2{\beta }_{\Psi }^{\text{'}})\le a\left(l^{\Phi }\right)\le \frac{2}{{\theta }_{\Phi }^0}$.

$\alpha$-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.

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 $D$ be a nonempty compact subset of a Banach space $X$ and denote by $S\left(X\right)$ the family of all nonempty bounded closed convex subsets of $X$. We endow $S\left(X\right)$ with the Hausdorff metric and show that there exists a set $ℱ\subset S\left(X\right)$ such that its complement $S\left(X\right)\setminus ℱ$ is $\sigma$-porous and such that for each $A\in ℱ$ and each $\tilde{x}\in D$, the set of solutions of the best approximation problem $\parallel \tilde{x}-z\parallel \to min$, $z\in A$, is nonempty and compact, and each minimizing sequence has a convergent subsequence.