Invariance of Domain

This lecture explores the invariance of domain theorem, which states that a subset of Rn homeomorphic to an open subset of Rn is open itself. The instructor proves this theorem by showing that the image of a continuous map from an open subset to Rn is open, using the one-point compactification of Rn. The lecture also discusses the implications of this theorem, such as the surjectivity of embeddings and homeomorphisms. The invariance of domain theorem, originally proved by Breuer in the early 20th century, is a fundamental result in topology with applications in geometry.