Editing Geometric topology (section)

===Local flatness===
{{main|Local flatness}}
[[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]].

Suppose a ''d'' dimensional manifold ''N'' is embedded into an ''n'' dimensional manifold ''M'' (where ''d'' &lt; ''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>.