Let \(\exp(X)\) denote the exponential space of a topological space \(X\) introduced by Vietoris. The paper studies the subspaces of \(\exp(\mathbb{N})\), where \(\mathbb{N}\) is the discrete space of the natural numbers. The following result in this paper is highlighted. If the metrizability number of \(\exp(X)\) is countable, then \(X\) and \(\exp(X)\) must be compact and metrizable. This result is built upon many other interesting results about \(\exp(\mathbb{N})\) proved in the paper.