The Hartman--Mycielski construction in topological MV-algebras

Abstract

Recently, topological MV-algebras have been investigated by several mathematicians. In this paper, we mainly show that for every Hausdorff topological MV-algebra $A$, there exists a natural topological isomorphism $i_A:A\to A^{\bullet }$ of $A$ onto a closed subalgebra of the pathwise connected, locally pathwise connected topological MV-algebra $A^{\bullet }$. Furthermore, we show that there is an extension to a bounded continuous function on the MV-algebra $A^{\bullet }$ for each continuous real-valued bounded function on a topological MV-algebra $A$. Finally, we prove that if $\phi :A_1\to A_2$ is a continuous homomorphism of topological MV-algebras, then $\phi$ admits a natural extension to a continuous homomorphism ${\phi }^{\bullet }:A_1^{\bullet }\to A_2^{\bullet }$; in addition, if $\phi$ is open and onto, then so is ${\phi }^{\bullet }$.