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In this note we prove a Birkhoff type transitivity theorem for continuousmaps acting on non-separable completely metrizable spaces and we give someapplications for dynamics of bounded linear operators acting on complexFr\'{e}chet spaces. Among them we show that any positive power and anyunimodular multiple of a topologically transitive linear operator istopologically transitive, generalizing similar results of S.I. Ansari and F.Le\'{o}n-Saavedra V. M\"{u}ller for hypercyclic operators.

A Birkhoff type transitivity theorem for non-separable completely metrizable spaces with applications to Linear Dynamics