Editing Embedding (section)

====Related definitions====

If the domain of a function <math>f : X \to Y</math> is a [[topological space]] then the function is said to be ''{{visible anchor|locally injective at a point}}'' if there exists some [[Neighbourhood (mathematics)|neighborhood]] <math>U</math> of this point such that the restriction <math>f\big\vert_U : U \to Y</math> is injective. It is called ''{{visible anchor|locally injective}}'' if it is locally injective around every point of its domain. Similarly, a ''{{visible anchor|local topological embedding|text=local (topological, resp. smooth) embedding}}'' is a function for which every point in its domain has some neighborhood to which its restriction is a (topological, resp. smooth) embedding. 

Every injective function is locally injective but not conversely. [[Local diffeomorphism]]s, [[local homeomorphism]]s, and smooth [[Immersion (mathematics)|immersion]]s are all locally injective functions that are not necessarily injective. The [[inverse function theorem]] gives a sufficient condition for a continuously differentiable function to be (among other things) locally injective. Every [[Fiber (mathematics)|fiber]] of a locally injective function <math>f : X \to Y</math> is necessarily a [[Discrete space|discrete subspace]] of its [[Domain of a function|domain]] <math>X.</math>