Mapping space

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A mapping space can be equipped with several topologies. A common one is the compact-open topology or the k-ification of it. 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.)

Here is another common one. We have:

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

given by ${\displaystyle f\mapsto \Gamma _{f}=}$ the graph of ${\displaystyle f}$. Then we can give ${\displaystyle \operatorname {Map} (X,Y)}$ the Whitney topology (also called the fine topology or the strong topology) where a basic open set consists of those ${\displaystyle g}$ such that ${\displaystyle \Gamma _{g}\subset U}$ for some open subset ${\displaystyle U\subset X\times Y}$.^[1][2] The compact-open topology does not handle a behavior at infinity well and so sometimes the Whitney topology is used instead.

If ${\displaystyle X}$ is a paracompact^[3] and ${\displaystyle Y}$ is a metric space, then the Whitney topology has a basic open set of the form

${\displaystyle B(f,\epsilon ):=\{g\mid d(g(x),f(x))<\epsilon (x)\}}$

for some ${\displaystyle f\in \operatorname {Map} (X,Y)}$ and some continuous function ${\displaystyle \epsilon :X\to \mathbb {R} _{>0}}$. If, moreover, ${\displaystyle Y}$ is complete, then we have the following important fact:

Let ${\displaystyle Q\subset \operatorname {Map} (X,Y)}$ be a subset such that every uniform limit of a sequence in ${\displaystyle Q}$, if any, is in ${\displaystyle Q}$. Then ${\displaystyle Q}$ is a Baire space.^[4]

This is proved by the same way Baire's category theorem is proved except we use the above family-version of a ball.

For manifolds ${\displaystyle M,N}$, there is the subset ${\displaystyle {\mathcal {C}}^{r}(M,N)\subset \operatorname {Map} (M,N)}$ 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 ${\displaystyle 1\leq s\leq \infty ,0\leq r<s}$.^[5]

See also: Grauert's approximation theorem

A basic result here is a theorem of Milnor which says that the mapping space ${\displaystyle \operatorname {Map} (X,Y)}$ has the homotopy type of a CW-complex if ${\displaystyle X}$ is a compact Hausdorff space and ${\displaystyle Y}$ has the homotopy type of a CW-complex.^[6]