Abstract: | We investigate observational constraints on the running vacuum model (RVM) of \begin{document}$\Lambda=3\nu (H^{2}+K/a^2)+c_0$\end{document}![]() in a spatially curved universe, where \begin{document}$\nu$\end{document}![]() is the model parameter, \begin{document}$K$\end{document}![]() corresponds to the spatial curvature constant, \begin{document}$a$\end{document}![]() represents the scalar factor, and \begin{document}$c_{0}$\end{document}![]() is a constant defined by the boundary conditions. We study the CMB power spectra with several sets of \begin{document}$\nu$\end{document}![]() and \begin{document}$K$\end{document}![]() in the RVM. By fitting the cosmological data, we find that the best fitted \begin{document}$\chi^2$\end{document}![]() value for RVM is slightly smaller than that of \begin{document}$\Lambda$\end{document}![]() CDM in the non-flat universe, along with the constraints of \begin{document}$\nu\leqslant O(10^{-4})$\end{document}![]() (68% C.L.) and \begin{document}$|\Omega_K=-K/(aH)^2|\leqslant O(10^{-2})$\end{document}![]() (95% C.L.). In particular, our results favor the open universe in both \begin{document}$\Lambda$\end{document}![]() CDM and RVM. In addition, we show that the cosmological constraints of \begin{document}$\Sigma m_{\nu}=0.256^{+0.224}_{-0.234}$\end{document}![]() (RVM) and \begin{document}$\Sigma m_{\nu}=0.257^{+0.219}_{-0.234}$\end{document}![]() (\begin{document}$\Lambda$\end{document}![]() CDM) at 95% C.L. for the neutrino mass sum are relaxed in both models in the spatially curved universe. |