AbstractLet Ω1, Ω2be open subsets of ℝand ℝ, respectively, and let A(Ω1) denote the space of real analytic functions on Ω1. We prove a Glaeser type theorem by characterizing when a composition operatorCφ: A(Ω1) → A(Ω2),Cφ(f) ≔f∘φ, is a topological embedding. Using this result we characterize when A(Ω1) can be embedded topologically into A(Ω2) as a locally convex space or as a topological algebra. We also characterize LB–subspaces and Fréchet subspaces of A(Ω1). In particular, it follows that if A(Ω1) and A(Ω2) are isomorphic as locally convex spaces, thend1=d2.