We define path-connected topological spaces and show that many of the properties of connected spaces also hold for path-connectedness. Moreover, we prove that path-connectedness implies connectedness, and see a counterexample to the converse statement.
00:00 Introduction
00:25 Motivation
02:10 Definition: Path
03:42 Definition: Path-Connectedness
04:25 Continuous images preserve path-connectedness
10:11 Unions with common point preserve path-connectedness
18:51 Finite products preserve path-connectedness
24:15 Quotients preserve path-connectedness
26:00 Thm: Path-connectedness implies connectedness
31:56 Examples of path-connected spaces
37:28 A connected space which is not path-connected
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