In mathematics, especially in

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

, 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

h: I \to \operatorname(X, Y)

in the mapping space is exactly a

homotopy In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. ...

Topologies

A mapping space can be equipped with several topologies. A common one is the

compact-open topology In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory ...

. Typically, there is then the adjoint relation :

\operatorname(X \times Y, Z) \simeq \operatorname(X, \operatorname(Y, Z))

and thus

\operatorname

is an analog of the

Hom functor In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between object (category theory), objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applicati ...

. (For pathological spaces, this relation may fail.)

Smooth mappings

For manifolds

M, N

, there is the subspace

\mathcal^r(M, N) \subset \operatorname(M, N)

that consists of all the

\mathcal^r

-smooth maps from

M

N

. It can be equipped with the weak or strong topology. A basic approximation theorem says that

\mathcal_W^s(M, N)

is dense in

\mathcal_S^r(M, N)

for

1 \le s \le \infty, 0 \le r < s

References

* * {{topology-stub Algebraic topology