Continuous functions on primal topological spaces induced by group actions

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<p>If $ G $ is a group acting on a set $ X $, then for any $ a\in G $, the restriction $ \phi_a:X\to X $ of the action to $ a $ induces a topology $ \tau_a $ for $ X $, called the primal topology induced by $ \phi_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_{ \phi_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.</p>