An answer to a problem on topologies of function spaces on metric measure spaces

Abstract

Let X be a metric measure space. In the paper K. Koshino (2020) [6], it was proved that if X satisfies some conditions, then the $L^p$-space $L^p\left(X\right)$ on X is homeomorphic to the product space s of countably infinitely many open intervals $(-1,1)$, and the subspace $C_u\left(X\right)$ of $L^p\left(X\right)$ consisting of uniformly continuous maps is also homeomorphic to the subspace $c_0$ of s consisting of sequences converging to 0. Then it was asked whether or not the pair $\left(L^p\right(X),C_u(X\left)\right)$ is homeomorphic to $(s,c_0)$. In this note, we will present some examples of metric measure spaces X in the both cases where those pairs are homeomorphic and not, and show that they are not homeomorphic if X is a Euclidean space or its cube with the usual metric and the Lebesgue measure.