# ${\mathrm{\pi \x9d\x92\x9e}}^{r}$ topologies

The ${\mathrm{\pi \x9d\x92\x9e}}^{r}$ Whitney (or strong) topology^{} is a topology
assigned to the space ${\mathrm{\pi \x9d\x92\x9e}}^{r}\beta \x81\u2019(M,N)$ of mappings from
a ${\mathrm{\pi \x9d\x92\x9e}}^{r}$ manifold^{} $M$ to a ${\mathrm{\pi \x9d\x92\x9e}}^{r}$ manifold $N$ having
$r$ continuous^{} derivatives^{} . It gives a notion of proximity
of two ${\mathrm{\pi \x9d\x92\x9e}}^{r}$ mappings, and it allows us to speak of βrobustnessβ
of properties of a mapping. For example, the
property of being an embedding^{} is robust: if $f:M\beta \x86\x92N$
is a ${\mathrm{\pi \x9d\x92\x9e}}^{r}$ embedding, then there is a strong ${\mathrm{\pi \x9d\x92\x9e}}^{r}$
neighborhood of $f$ in which any ${\mathrm{\pi \x9d\x92\x9e}}^{r}$ mapping $g:M\beta \x86\x92N$
is an embedding.

Given a locally finite^{} atlas $\{({U}_{i},{\mathrm{{\rm O}\x95}}_{i}):i\beta \x88\x88I\}$ and compact sets
${K}_{i}\beta \x8a\x82{U}_{i}$ such that there are charts
$\{({V}_{i},{\mathrm{{\rm O}\x88}}_{i}):i\beta \x88\x88I\}$ of $N$ for which
$f\beta \x81\u2019({K}_{i})\beta \x8a\x82{V}_{i}$ for all $i\beta \x88\x88I$, and given a sequence
$\{{\mathrm{{\rm O}\u0385}}_{i}>0:i\beta \x88\x88I\}$, we define the basic neighborhood

$${\mathrm{\pi \x9d\x92\xb0}}^{r}\beta \x81\u2019(f,\mathrm{{\rm O}\x95},\mathrm{{\rm O}\x88},\{{K}_{i}:i\beta \x88\x88I\},\{{\mathrm{{\rm O}\u0385}}_{i}:i\beta \x88\x88I\})$$ |

as the set of ${C}^{r}$ mappings $g:M\beta \x86\x92N$ such that for all $i\beta \x88\x88I$ we have $g\beta \x81\u2019({K}_{i})\beta \x8a\x82{V}_{i}$ and

$$ |

That is, those maps $g$ that are close to $f$ and have their first $r$ derivatives close to the respective first $r$-th derivatives of $f$, in local coordinates. It can be checked that the set of all such neighborhoods forms a basis for a topology, which we call the Whitney or strong ${\mathrm{\pi \x9d\x92\x9e}}^{r}$ topology of ${\mathrm{\pi \x9d\x92\x9e}}^{r}\beta \x81\u2019(M,N)$.

The weak ${\mathrm{\pi \x9d\x92\x9e}}^{r}$ topology, or ${\mathrm{\pi \x9d\x92\x9e}}^{r}$ compact-open topology^{}, is defined
in the same fashion but instead of choosing
$\{({U}_{i},{\mathrm{{\rm O}\x95}}_{i}):i\beta \x88\x88I\}$ to be a locally finite atlas for $M$,
we require it to be an arbitrary *finite* family of charts
(possibly not covering $M$).

The space ${\mathrm{\pi \x9d\x92\x9e}}^{r}\beta \x81\u2019(M,N)$ with the weak or strong topologies is denoted by ${\mathrm{\pi \x9d\x92\x9e}}_{W}^{r}\beta \x81\u2019(M,N)$ and ${\mathrm{\pi \x9d\x92\x9e}}_{S}^{r}\beta \x81\u2019(M,N)$, respectively.

We have that ${\mathrm{\pi \x9d\x92\x9e}}_{W}^{r}\beta \x81\u2019(M,N)$ is always metrizable (with a complete metric)
and separable^{}. On the other hand, ${\mathrm{\pi \x9d\x92\x9e}}_{S}^{r}\beta \x81\u2019(M,N)$ is not even first countable (thus, not metrizable) when $M$ is not compact; however, it is a Baire space^{}. When $M$ is compact, the weak and strong topologies coincide.

Title | ${\mathrm{\pi \x9d\x92\x9e}}^{r}$ topologies |
---|---|

Canonical name | mathcalCrTopologies |

Date of creation | 2013-03-22 14:08:27 |

Last modified on | 2013-03-22 14:08:27 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 5 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 57R12 |

Synonym | Whitney topology |

Synonym | compact-open ${\mathrm{\pi \x9d\x92\x9e}}^{r}$ topology |

Synonym | weak ${\mathrm{\pi \x9d\x92\x9e}}^{r}$ topology |

Synonym | strong $\text{mathcak}\beta \x81\u2019{C}^{r}$ topology |

Related topic | ConjectureApproximationTheoremHoldsForWhitneyCrMNSpaces |

Related topic | ApproximationTheoremAppliedToWhitneyCrMNSpaces |