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Let K be a closed Lie subgroup of the unitary group U(n) acting by automorphisms on the (2n+1)-dimensional Heisenberg group $H_n$. We say that $(K,H_n)$ is a Gelfand pair when the set $L_K^1\left(H_n\right)$ of integrable K-invariant functions on $H_n$ is an abelian convolution algebra. In this case, the Gelfand space (or spectrum) for $L_K^1\left(H_n\right)$ can be identified with the set $\Delta (K,H_n)$ of bounded K-spherical functions on $H_n$. In this paper, we study the natural topology on $\Delta (K,H_n)$ given by uniform convergence on compact subsets in $H_n$. We show that $\Delta (K,H_n)$ is a complete...