A study on symmetric products of generalized metric spaces

Abstract

We study the relation between a space X satisfying certain generalized metric properties (for example, open (G), point-countable base, Collins–Roscoe property, semi-stratifiable, k-semistratifiable, semi-metrizable, scattered, point-countable cs-network, every compact set is metrizable) and its n-fold symmetric product Fn(X) satisfying the same properties. We also show that if X is an M1-space then F(X) is an M1-space, where F(X) is the hyperspace of finite subsets of X. A space X is a paracompact p-space if and only if its 2-fold symmetric product F2(X) is a paracompact p-space. A Tychonoff space X is a Lindelöf Σ-space if and only if its 2-fold symmetric product F2(X) is a Lindelöf Σ-space.