Lecture

Invariance of Domain

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In course

MATH-323: Topology III - Homology

Homology is one of the most important tools to study topological spaces and it plays an important role in many fields of mathematics. The aim of this course is to introduce this notion, understand its properties and learn how to compute it. There will be m ...

Description

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.

Instructor

Leonid Monin

Teaching & PhD Courses Algebra III - rings and fields MATH-215 This is an introductory course in ring and field theory. Analysis I (English) MATH-101(en) We study the fundamental concepts of analysis, calculus and the integral of real-valued functions of a real variable. Topology III - Homology MATH-323 Homology is one of the most important tools to study topological spaces and it plays an important role in many fields of mathematics. The aim of this course is

Official source

Ontological neighbourhood

Mathematics

Topology: General topology

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