Topology Lecture 06: Open / Closed Maps

We define open maps, closed maps, and local homeomorphisms. These can be seen as different ways of weakening the concept of homeomorphism. We then show that when the maps in question are bijective, the above notions coincide.

00:00 Introduction
00:24 Definition: Open Map
02:35 Definition: Closed Map
04:50 Prop: Bijective open / closed maps are homeomorphisms
18:59 Definition: Local Homeomorphism
21:33 Example: Real line covers circle
25:13 Prop: Bijective local homeomorphism is a homeomorphism

This lecture follows Lee's "Introduction to topological manifolds", chapter 2.

A playlist with all the videos in this series can be found here:

[eamon_concannon]

30:06 I think that f has also to be shown to be continuous to conclude that f is an open map.

[eamon_concannon]

17:43 How do we know that f is continuous? You just showed that a closed bijective funtion f has a continuous inverse.

[ian]

i guess that f need to be continuos to make a homeomorphism