We construct a ‘counterexample’ to Łoś’ theorem in the ordered Mostowski model for set theory ZFA.

A mapping $f$:$X\rightarrow Y$ between continua $X$ and $Y$ called an MO-mapping provided that it can be represented as the composition of two mappings, $f_{1}$: $X\rightarrow Z$ and $f_{2}$:$Z\rightarrow Y$, such that $f_{1}$ is open and $f_{2}$ is monotone. Induced MO-mappings, $2^{f}$ and $C(f)$ between hyperspaces are studied. In particular an example is constructed of an open mapping $f$: $[0,1]\rightarrow[0,1]$ for which $C(f)$ is not an MO-mapping. This answers two questions asked by H. Hosokawa.

In this paper, we give a classification of real hypersurfaces of a complex projective space $CP^{n}$ satisfying a pointwise nullity condition for the structure vector field $\xi$ i.e., $R(X, Y)\xi=k\{\eta(Y)X-\eta(X)Y\}$, $k$ is a function, and further we prove a local structure theorem of real hypersurfaces of $CP^{n}$ which satisfies $R(X, A\xi)\xi=k\{\eta(A\xi)X-\eta(X)A\xi\}$. The motivation of the present paper is a well-known fact that $CP^{n}$ does not admit a real hypersurface of constant curvature.

Some basic properties of graded Azumaya algebras are given systematically. Here, the grading is indexed by an arbitrary additive group and the notion of a graded Azumaya algebra is relative to a choice of a commutation factor. We also establish two exact sequences which generalize the Skolem-Noether theorem.