In this paper we study the topological and metric rigidity of hypersurfaces in ${\mathbb H}^{n+1}$, the $(n+1)$-dimensional hyperbolic space of sectional curvature $-1$. We find conditions to ensure a complete connected oriented hypersurface in ${\mathbb H}^{n+1}$ to be diffeomorphic to a Euclidean sphere. We also give sufficient conditions for a complete connected oriented closed hypersurface with constant norm of the second fundamental form to be totally umbilic.
Pointfree formulas for three kinds of separating points for closed sets by maps are given. These formulas allow controlling the amount of factors of the target product space so that it does not exceed the weight of the embeddable space. In literature, the question of how many factors of the target product are needed for the embedding has only been considered for specific spaces. Our approach is algebraic in character and can thus be viewed as a contribution to Kuratowski's topological calculus.
For an order embedding $G\overset{h}{\rightarrow }{\rightarrow }\Gamma $ of a partly ordered group $G$ into an $l$-group $\Gamma $ a topology $\mathcal T_{\widehat{W}}$ is introduced on $\Gamma $ which is defined by a family of valuations $W$ on $G$. Some density properties of sets $h(G)$, $h(X_t)$ and $(h(X_t)\setminus \lbrace h(g_1),\dots ,h(g_n)\rbrace )$ ($X_t$ being $t$-ideals in $G$) in the topological space $(\Gamma ,\mathcal T_{\widehat{W}})$ are then investigated, each of them being equivalent to the statement that $h$ is a strong theory of quasi-divisors.