We study countable partitions for measurable maps on measure spaces such that, for every point x, the set of points with the same itinerary as that of x is negligible. We prove in nonatomic probability spaces that every strong generator (Parry, W., Aperiodic transformations and generators, J. London Math. Soc. 43 (1968), 191–194) satisfies this property (but not conversely). In addition, measurable maps carrying partitions with this property are aperiodic and their corresponding spaces are nonatomic. From this we obtain a characterization of nonsingular countable-to-one mappings with these partitions on nonatomic Lebesgue probability spaces as those having strong generators. Furthermore, maps carrying these partitions include ergodic measure-preserving ones with positive entropy on probability spaces (thus extending the result in Cadre, B., Jacob, P., On pairwise sensitivity, J. Math. Anal. Appl. 309 (2005), 375–382). Some applications are given.
Lee, Kim and Suh (2012) gave a characterization for real hypersurfaces $M$ of Type ${\rm (A)}$ in complex two plane Grassmannians $G_2({\mathbb C}^{m+2})$ with a commuting condition between the shape operator $A$ and the structure tensors $\phi $ and $\phi _{1}$ for $M$ in $G_2({\mathbb C}^{m+2})$. Motivated by this geometrical notion, in this paper we consider a new commuting condition in relation to the shape operator $A$ and a new operator $\phi \phi _{1}$ induced by two structure tensors $\phi $ and $\phi _{1}$. That is, this commuting shape operator is given by $\phi \phi _{1} A = A \phi \phi _{1}$. Using this condition, we prove that $M$ is locally congruent to a tube of radius $r$ over a totally geodesic $G_2({\mathbb C}^{m+1})$ in $G_2({\mathbb C}^{m+2})$.
We explore (weak) continuity properties of group operations. For this purpose, the Novak number and developability number are applied. It is shown that if $(G, \cdot ,\tau )$ is a regular right (left) semitopological group with $\mathop{{\rm dev}}(G)<\mathop{{\rm Nov}}(G)$ such that all left (right) translations are feebly continuous, then $(G,\cdot ,\tau )$ is a topological group. This extends several results in literature.