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local homeomorphism∗
joking†
2013-03-22 2:47:19
Definition. Let X and Y be topological spaces. Continuous map f : X →
Y is said to be locally invertible in x ∈ X iff there exist open subsets U ⊆ X
and V ⊆ Y such that x ∈ U , f (x) ∈ V and the restriction
f :U →V
is a homeomorphism. If f is locally invertible in every point of X, then f is
called a local homeomorphism.
Examples. Of course every homeomorphism is a local homeomorphism,
but the converse is not true. For example, let f : C → C be an exponential
function, i.e. f (z) = ez . Then f is a local homeomorphism, but it is not a
homeorphism (indeed, f (z) = f (z + 2πi) for any z ∈ C).
One of the most important theorem of differential calculus (i.e. inverse
function theorem) states, that if f : M → N is a C 1 -map between C 1 -manifolds
such that Tx f : Tx M → Tf (x) N is a linear isomorphism for a given x ∈ M , then
f is locally invertible in x (in this case the local inverse is even a C 1 -map).
∗ hLocalHomeomorphismi created: h2013-03-2i by: hjokingi version: h41743i Privacy
setting: h1i hDefinitioni h54C05i
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