We define connected topological spaces, present two characterizations, several properties, and finally classify all connected subsets of the real line.
00:00 Introduction
00:17 Motivation
02:04 Definition: Connected Space
04:24 Examples of disconnected spaces
12:32 Examples of connected spaces
13:34 Prop: Only emptyset and X are clopen in connected X.
19:42 Prop: Connected spaces are not disjoint union of smaller spaces
24:07 Prop: Continuous images of connected space are connected.
32:47 Prop: Connected subsets cannot be shared between open disjoint sets
38:16 Prop: Unions of connected spaces that share a point are connected
42:27 Prop: Finite products of connected spaces are connected
48:54 Prop: Quotients of connected spaces are connected
50:50 Prop: The nonempty connected subsets of R are points and intervals
1:15:15 Prop: Generalized intermediate value theorem
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