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Let B be an F-algebra and i be a commutative semisimple F-algebra such that the spectrum of A contains no isolated points. We prove that any homomorphism of B onto A is necessarily continuous. Let A be a commutative semisimple algebra. We prove that there is at most one topology with respect to which A is an F-algebra.

Let A be a commutative semisimple F-algebra with identity, and let D 0 , D 1 , ⋯ {D_0},{D_1}, \cdots be a system of derivations from A into the algebra of all continuous functions on the spectrum of A. It is shown that the transformations D 0 , D 1 , ⋯ {D_0},{D_1}, \cdots are necessarily continuous. This result is used to obtain a characterization of derivations on Hol ( Ω ) {\text {Hol}}(\Omega ) where Ω \Omega is an open polynomially convex subset of C n {C^n} .