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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 ${\displaystyle h:I\to \operatorname {Map} (X,Y)}$ in the mapping space is exactly a homotopy.

Topologies

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

${\displaystyle \operatorname {Map} (X\times Y,Z)\simeq \operatorname {Map} (X,\operatorname {Map} (Y,Z))}$

and thus ${\displaystyle \operatorname {Map} }$ is an analog of the Hom functor. (For pathological spaces, this relation may fail.)

Smooth mappings

For manifolds ${\displaystyle M,N}$, there is the subspace ${\displaystyle {\mathcal {C}}^{r}(M,N)\subset \operatorname {Map} (M,N)}$ that consists of all the ${\displaystyle {\mathcal {C}}^{r}}$-smooth maps from ${\displaystyle M}$ to ${\displaystyle N}$. It can be equipped with the weak or strong topology.

A basic approximation theorem says that ${\displaystyle {\mathcal {C}}_{W}^{s}(M,N)}$ is dense in ${\displaystyle {\mathcal {C}}_{S}^{r}(M,N)}$ for ${\displaystyle 1\leq s\leq \infty ,0\leq r<s}$.

References

1. Hirsch, Ch. 2., § 2., Theorem 2.6. harvnb error: no target: CITEREFHirsch (help)

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

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