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.

A mapping space can be equipped with several topologies. A common one is the compact-open topology. Typically, there is then the adjoint relation

and thus is an analog of the Hom functor. (For pathological spaces, this relation may fail.)

For manifolds , there is the subspace that consists of all the -smooth maps from to . It can be equipped with the weak or strong topology.

A basic approximation theorem says that is dense in for .^[1]

• Hirsch, Morris (1997). Differential Topology. Springer. ISBN 0-387-90148-5.
• Wall, C. T. C. (4 July 2016). Differential Topology. Cambridge University Press. ISBN 9781107153523.