Symmetric products and closed finite-to-one mappings

In this paper, we continue the study of the symmetric products of generalized metric spaces in [39]. We consider the topological properties P such that the n-fold symmetric product Fn(X) of a topological space X has the topological properties P if and only if the space X or the product Xn does for each or some n∈N. Depending on the operations under closed subspaces, finite products and closed finite-to-one mappings, two general stability theorems are obtained on symmetric products. We can apply the methods to unify and simplify the proofs of some old results in the literature and obtain some new results on symmetric products, list or prove 68 topological properties which satisfy the general stability theorems, and give answers to Questions 3.6 and 3.35 in [39].