From the point of view of musical performance, gestures are the movements of the body of the performer when playing an instrument. This vague idea can be modeled mathematically, by mixing category theory and topology, giving rise to the definition of a topological gesture with a given skeleton and body in a topological space. The skeleton represents the abstract configuration of the body's limbs and the topological space is a generalization of the three-dimensional space where the body's movements are usually modeled. The collection of all gestures with the same skeleton and body in a fixed space has a canonical topology, yielding a space of gestures. This article intends to show that the space of gestures Γ@X is homeomorphic to the function space Top(|Γ|,X), endowed with the compact-open topology. The topology of this space is the most natural choice for a space of functions, in the sense that it is related to the universal property of exponentials in the category of topological spaces. In particular, when the skeleton has a suitable property of finiteness, we show that the function space becomes a true exponential.