Open main menu

Topospaces β

Tame submanifold

This article defines a property of a submanifold inside a manifold

Definition

Let M be a manifold of dimension m and N a submanifold of dimension n. Then N is termed tame in M if for every point x \in N, there exists a neighbourhood U of x in M such that the pair (U, U \cap N) is homeomorphic to the pair (\R^m,\R^n) where \R^n is viewed as a linear subspace of \R^m.

Another way of saying this is that the local codimension at each point, equals the codimension of the submanifold as a whole.

Facts

An example of a submanifold which is not tame is the Alexander horned sphere in \R^3.