Quotient homomorphisms of Topological MV-Algebras and Applications

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 $q:A⟶A\mathrm{/}I$, where $A\mathrm{/}I$ is endowed with the quotient topology, is a continuous open quotient map and $A\mathrm{/}I$ 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, $\sigma$-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.