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 local flatness ： ウィキペディア英語版
local flatness
In topology, a branch of mathematics, local flatness is a property of a submanifold in a topological manifold of larger dimension. In the category of topological manifolds, locally flat submanifolds play a role similar to that of embedded submanifolds in the category of smooth manifolds.
Suppose a ''d'' dimensional manifold ''N'' is embedded into an ''n'' dimensional manifold ''M'' (where ''d'' < ''n''). If $x \in N,$ we say ''N'' is locally flat at ''x'' if there is a neighborhood $U \subset M$ of ''x'' such that the topological pair $\left(U, U\cap N\right)$ is homeomorphic to the pair $\left(\mathbb^n,\mathbb^d\right)$, with a standard inclusion of $\mathbb^d$ as a subspace of $\mathbb^n$. That is, there exists a homeomorphism $U\to R^n$ such that the image of $U\cap N$ coincides with $\mathbb^d$.
The above definition assumes that, if ''M'' has a boundary, ''x'' is not a boundary point of ''M''. If ''x'' is a point on the boundary of ''M'' then the definition is modified as follows. We say that ''N'' is locally flat at a boundary point ''x'' of ''M'' if there is a neighborhood $U\subset M$ of ''x'' such that the topological pair $\left(U, U\cap N\right)$ is homeomorphic to the pair $\left(\mathbb^n_+,\mathbb^d\right)$, where $\mathbb^n_+$ is a standard half-space and $\mathbb^d$ is included as a standard subspace of its boundary. In more detail, we can set
$\mathbb^n_+ = \$ and $\mathbb^d = \=\cdots=y_n=0\\right\}$.
We call ''N'' locally flat in ''M'' if ''N'' is locally flat at every point. Similarly, a map $\chi\colon N\to M$ is called locally flat, even if it is not an embedding, if every ''x'' in ''N'' has a neighborhood ''U'' whose image $\chi\left(U\right)$ is locally flat in ''M''.
Local flatness of an embedding implies strong properties not shared by all embeddings. Brown (1962) proved that if ''d'' = ''n'' − 1, then ''N'' is collared; that is, it has a neighborhood which is homeomorphic to ''N'' × () with ''N'' itself corresponding to ''N'' × 1/2 (if ''N'' is in the interior of ''M'') or ''N'' × 0 (if ''N'' is in the boundary of ''M'').