Let Q be the space of all rational numbers and (X, r) be a topological space where X is countably infinite. Here we prove that (1) r is the join of two topologies on X both homeomorphic to Q if and only if τ is non-compact and metrizable, and (2) r is the join of topologies on X each homeomorphic to Q if and only if r is Tychonoff and noncompact. 1980 Mathematics subject classification (Amer. Math. Soc.) (1985 Revision): 54 E 35. © 1989, Australian Mathematical Society. All rights reserved.