We introduce the concepts of components and path-components before turning to the idea of local (path-) connectivity. The main result is that in a locally path-connected space (e.g. in topological manifolds), the notions of connectedness and path-connectedness coincide.
00:00 Introduction
00:20 Definition: Components
02:43 Examples of components
09:37 Components partition the space
18:35 Properties of components
23:45 Definition: Path-components
28:24 Definition: Local (Path-) Connectedness
30:27 Examples of local connectedness
37:27 Connectedness does not imply local connectedness
43:37 Properties of locally connected spaces
49:40 Properties of locally path-connected spaces
This lecture follows Lee's "Introduction to topological manifolds", chapter 4.
A playlist with all the videos in this series can be found here:
https://www.youtube.com/playlist?list=PLd8NbPjkXPliJunBhtDNMuFsnZPeHpm-0