Continuous functions on primal topological spaces induced by group actions - Luis Fernando Mejías; Jorge Vielma; Elvis Aponte; Lourival Rodrigues De Lima (AIMS Mathematics)
If $ G $ is a group acting on a set $ X $, then for any $ a\in G $, the restriction $ φ_a:X→ X $ of the action to $ a $ induces a topology $ τ_a $ for $ X $, called the primal topology induced by $ φ_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, τ_{ φ_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.