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| In [[topology]], a branch of [[mathematics]], '''local flatness''' is a property of a [[submanifold]] in a [[topological manifold]] of larger [[dimension]]. In the [[Category (mathematics)|category]] of topological manifolds, locally flat submanifolds play a role similar to that of [[Submanifold#Embedded submanifolds|embedded submanifolds]] in the category of [[smooth manifolds]].
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| Suppose a ''d'' dimensional manifold ''N'' is embedded into an ''n'' dimensional manifold ''M'' (where ''d'' < ''n''). If <math>x \in N,</math> we say ''N'' is '''locally flat''' at ''x'' if there is a neighborhood <math> U \subset M</math> of ''x'' such that the [[topological pair]] <math>(U, U\cap N)</math> is [[homeomorphic]] to the pair <math>(\mathbb{R}^n,\mathbb{R}^d)</math>, with a standard inclusion of <math>\mathbb{R}^d</math> as a subspace of <math>\mathbb{R}^n</math>. That is, there exists a homeomorphism <math>U\to R^n</math> such that the [[image (mathematics)|image]] of <math>U\cap N</math> coincides with <math>\mathbb{R}^d</math>.
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| The above definition assumes that, if ''M'' has a [[Boundary (topology)|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 <math>U\subset M</math> of ''x'' such that the topological pair <math>(U, U\cap N)</math> is homeomorphic to the pair <math>(\mathbb{R}^n_+,\mathbb{R}^d)</math>, where <math>\mathbb{R}^n_+</math> is a standard [[Half-space (geometry)|half-space]] and <math>\mathbb{R}^d</math> is included as a standard subspace of its boundary. In more detail, we can set
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| <math>\mathbb{R}^n_+ = \{y \in \mathbb{R}^n\colon y_n \ge 0\}</math> and <math>\mathbb{R}^d = \{y \in \mathbb{R}^n\colon y_{d+1}=\cdots=y_n=0\}</math>.
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| We call ''N'' '''locally flat''' in ''M'' if ''N'' is locally flat at every point. Similarly, a map <math>\chi\colon N\to M</math> is called '''locally flat''', even if it is not an embedding, if every ''x'' in ''N'' has a neighborhood ''U'' whose image <math>\chi(U)</math> is locally flat in ''M''.
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| 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'' × [0,1] 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'').
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| ==See also==
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| *[[Neat submanifold]]
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| ==References==
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| * Brown, Morton (1962), Locally flat imbeddings of topological manifolds. ''Annals of Mathematics'', Second series, Vol. 75 (1962), pp. 331-341.
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| [[Category:Topology]]
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| [[Category:Geometric topology]]
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| {{topology-stub}}
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Hello, my name is Andrew and my spouse doesn't like it at all. She is truly fond of caving but she doesn't have the time lately. Kentucky is exactly where I've usually been living. Invoicing is what I do for a residing but I've usually needed my own business.
My web page: online reader