If $G$ is a group acting on a set $X$, then for any $a\in G$, the restriction ${\varphi }_a:X\to X$ of the action to $a$ induces a topology ${\tau }_a$ for $X$, called the primal topology induced by ${\varphi }_a$. First, we obtain a characterization of the normal subgroups in terms of the primal topologies. Later, we prove that some commutative relations among elements on the group $G$ determine the continuity of maps among different primal spaces $(X,{\tau }_{{\varphi }_x})$. In particular, we prove the continuity of some maps when $a,b,q\in G$ satisfy a quantum type relation, $ba=qab$, as is in the quaternion and Heisenberg groups.

In this paper, for a bounded linear operator defined on a complex Banach space of infinite dimension, we consider the set of isolated points in its approximate point spectrum, which are eigenvalues of finite multiplicity; this set can be equal to the spectrum of the operator but without its upper semi-Fredholm spectrum, and this relation or equality defines in the literature a new spectral property called the property $(\omega \pi )$ and is a generalization of the classical a-Weyl theorem. We establish some characterizations and consequences about the property $(\omega \pi )$, some with topological aspects. Furthermore, we study this property through the Riesz functional calculus. Part of the spectral structure of a linear operator verifying property $(\omega \pi )$ is described, obtaining some associated properties.