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The new $ν$-metric induces the classical gap topology

Let $\calA_+$ denote the set of Laplace transforms of complex Borel measures $μ$ on $[0,+\infty)$ such that $μ$ does not have a singular non-atomic part. In \cite{BalSas}, an extension of the classical $ν$-metric of Vinnicombe was given, which allowed one to address robust stabilization problems for unstable plants over $\calA_+$. In this article, we show that this new $ν$-metric gives a topology on unstable plants which coincides with the classical gap topology for unstable plants over $\calA_+$ with a single input and a single output.