Topic summary
Mapping space
In mathematics, especially in algebraic topology, the mapping space between two spaces is the space of all the (continuous) maps between them.
Viewing the set of all the maps as a space is useful because that allows for topological considerations. For example, a curve in the mapping space is exactly a homotopy between the starting point and the end point.
From the category theory point of view, a mapping space provides the internal Hom (i.e., hom that is also an object) in the category of spaces.