Homeomorphisms#
Definition: Homeomorphism
Let \((X, \tau_X)\) and \((Y,\tau_Y)\) be topological spaces.
A homeomorphism between \(X\) and \(Y\) is a continuous bijection \(f: X \to Y\) with a continuous inverse \(f^{-1}: Y \to X\).
Definition: Homeomorphic Spaces
Two topological spaces \((X, \tau_X)\) and \((Y, \tau_Y)\) are homeomorphic if there exists a homeomorphism between them.
Notation
Theorem: Composition of Homeomorphisms
Let \((X, \tau_X)\), \((Y, \tau_Y)\) and \((Z, \tau_Z)\) be topological spaces.
If \(f: X \to Y\) and \(g: Y \to Z\) are homeomorphisms, then their composition \(g \circ f: X \to Z\) is also a homeomorphism.
Proof
TODO
Local Homeomorphisms#
Definition: Local Homeomorphism
Let \((X, \tau_X)\) and \((Y,\tau_Y)\) be topological spaces.
A function \(f: X \to Y\) is a local homeomorphism from \((X, \tau_X)\) to \((Y, \tau_Y)\) if each \(x \in X\) has an open neighborhood \(O\) with an open image \(f(O)\) such that the restriction \(f\big|_O\) is a homeomorphism between the subspaces \(O\) and \(f(O)\).
Definition: Locally Homeomorphic Spaces
A topological space \((X, \tau_X)\) is locally homeomorphic to another topological space \((Y,\tau_Y)\) if there exists a local homeomorphism from \((X, \tau_X)\) to \((Y, \tau_Y)\).
Theorem: Homeomorphism \(\implies\) Local Homeomorphism
Every homeomorphism is also a local homeomorphism.
Proof
TODO