Topological properties of the space of $G$-symmetric degree

In this paper, we examine the weight, character, locally weak density, and metrizability of the space of $G$-symmetric degree. We proved that the mapping $\pi_{n,G}^{s}$ is open-closed, and the functor $SP_{G}^{n}$ preserves weight, net weight, character, local weak density, the Hausdorff property, regularity, completely regularity, metrizability, and connectedness.