We study those compactifications of a space such that every autohomeomorphism of the space can be continuously extended over the compactification. These are called H-compactifications. Van Douwen proved that there are exactly three H-compactifications of the real line. We prove that there exist only two H-compactifications of Euclidean spaces of higher dimension. Next we show that there are 26 H-compactifications of a countable sum of real lines and 11 H-compactifications of a countable sum of Euclidean...

This paper considers totally bounded quasi-uniformities and quasi-proximities for frames and shows that for a given quasi-proximity $◃$ on a frame $L$ there is a totally bounded quasi-uniformity on $L$ that is the coarsest quasi-uniformity, and the only totally bounded quasi-uniformity, that determines $◃$. The constructions due to B. Banaschewski and A. Pultr of the Cauchy spectrum $\psi L$ and the compactification $\Re L$ of a uniform frame $(L,𝐔)$ are meaningful for quasi-uniform frames. If $𝐔$ is a totally bounded quasi-uniformity...

Totally nonremote points in $\beta \mathbb{Q}$ are constructed. The number of these points is $2^{𝔠}$.

Let $G$ be a group, $\beta G$ be the Stone-Čech compactification of $G$ endowed with the structure of a right topological semigroup and $G^{*}=\beta G\setminus G$. Given any subset $A$ of $G$ and $p\in G^{*}$, we define the $p$-companion ${\Delta }_p\left(A\right)=A^{*}\cap Gp$ of $A$, and characterize the subsets with finite and discrete ultracompanions.

Let $F\subset C^{\ast }(X)$ be a vector sublattice over $ℝ$ which separates points from closed sets of $X$. The compactification $e_{F}X$ obtained by embedding $X$ in a real cube via the diagonal map, is different, in general, from the Wallman compactification $\omega (Z(F))$. In this paper, it is shown that there exists a lattice $F_z$ containing $F$ such that $\omega (Z(F))=\omega (Z(F_z))=e_{F}X$. In particular this implies that $\omega (Z(F))\ge e_{F}X$. Conditions in order to be $\omega (Z(F))=e_{F}X$ are given. Finally we prove that, if $\alpha X$ is a compactification of $X$ such that $C{l}_{\alpha X}(\alpha X\setminus X)$ is $0$-dimensional, then there is an algebra $A\subset C^{ast}(X)$ such...

In this paper we give a characterization of a separable metrizable space having a metrizable S-weakly infinite-dimensional compactification in terms of a special metric. Moreover, we give two characterizations of a separable metrizable space having a metrizable countable-dimensional compactification.