## Local Homeo(Diffeo)morphisms to Global Homeo(Diffeo)morphisms

**Point of Post: **In this post we discuss an important consequence of the inverse function theorem which relates local diffeomorphisms to global diffeomorphisms.

*Local to Global Diffeomorphism*

Let be open and . We call a *local diffeomorphism *if for every there exists a neighborhood of such that is open in and the map is a diffeomorphism. We define local homeomorphisms similarly. We note the obvious theorem

**Theorem: ***Let be open. Then, is a homeo(diffeo)morphism if and only if it’s a bijective local homeo(diffeo)morphism.*

Which is true by definition since differentiability is a local property. The important fact we’d like to note is the following:

**Theorem: ***Let be a local homeomorphism, then is an open map.*

**Proof: **Let be open. We know that for each there is neighborhood for which is open in and is a homeomorphism. Note then that

But, since is open in and is a homeomorphism we have that is open in , but since is open in this implies that is also open in and so is the union of open sets in and so trivially open.

So, what does this have to do with the inverse function theorem? Well, the following:

**Theorem: ***Let , with open, be such that the total derivative is invertible for each . Then, is a local diffeomorphism and so, in particular, open.*

**Proof: **This is an immediate consequence of the inverse function theorem which says that since is invertible has a neighborhood for which the restriction of to this neighborhood is a diffeomorphism.

**References:
**

1. Spivak, Michael. *Calculus on Manifolds; a Modern Approach to Classical Theorems of Advanced Calculus.* New York: W.A. Benjamin, 1965. Print.

2. Apostol, Tom M. *Mathematical Analysis*. Reading, MA: Addison-Wesley Pub., 1974. Print.

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