Quotient homomorphisms of Topological MV-Algebras and Applications

Abstract

For a topological group, the quotient map modulo a subgroup is open and the quotient map modulo a compact subgroup is perfect. In this paper we prove and develop the corresponding compact-ideal theory for topological \(MV\)-algebras. We show that if \(I\) is an ideal of a topological \(MV\)-algebra \(A\), then the natural quotient homomorphism \(q:A\longrightarrow A/I\), where \(A/I\) is endowed with the quotient topology, is a continuous open quotient map and \(A/I\) is again a topological \(MV\)-algebra. If, in addition, \(I\) is compact, then \(q\) is perfect. As applications, we study three-space phenomena in topological \(MV\)-algebras. Under compact-kernel hypotheses we prove three-space theorems for compactness, local compactness, \(σ\)-compactness, Lindelöfness and paracompactness under the separation hypotheses stated below. We also prove a first-countability three-space theorem for locally convex topological \(MV\)-algebras.