and are local homeomorphisms, then is also a local homeomorphism. Inclusion maps If is any subspace (where as usual, is equipped with the subspace topology induced by ) then the inclusion map is always a topological embedding. But it is a local homeomorphism if and only if is open in The subset being open in is essential for the inclusion map to be a local homeomorphism because the inclusion map of a non-open subset of yields a local homeomorphism (since it will not be an open map). The restriction of a function to a subset is equal to its composition with the inclusion map explicitly, Since the composition of two local homeomorphisms is a local homeomorphism, if and are local homomorphisms then so is Thus restrictions of local homeomorphisms to open subsets are local homeomorphisms. Invariance of domain Invariance of domain guarantees that if is a continuous injective map from an open subset of then is open in and is a homeomorphism. Consequently, a continuous map from an open subset will be a local homeomorphism if and only if it is a locally injective map (meaning that every point in has a neighborhood such that the restriction of to is injective). Local homeomorphisms in analysis It is shown in complex analysis that a complex analytic function (where is an open subset of the complex plane ) is a local homeomorphism precisely when the derivative is non-zero for all The function on an open disk around is not a local homeomorphism at when In that case is a point of "ramification" (intuitively, sheets come together there). Using the inverse function theorem one can show that a continuously differentiable function (where is an open subset of ) is a local homeomorphism if the derivative is an invertible linear map (invertible square matrix) for every (The converse is false, as shown by the local homeomorphism with ). An analogous condition can be formulated for maps between differentiable manifolds. Local homeomorphisms and fibers Suppose is a continuous open surjection between two Hausdorff second-countable spaces where is a Baire space and is a normal space. If every fiber of is a discrete subspace of (which is a necessary condition for to be a local homeomorphism) then is a -valued local homeomorphism on a dense open subset of To clarify this statement's conclusion, let be the (unique) largest open subset of such that is a local