Topology Lecture 11: Subspaces

We motivate and define the subspace topology. Then we show under what circumstances open (closed) sets in the subspace and larger space coincide. Finally, we discuss the characteristic property of the subspace topology and summarize some of its key properties.

00:00 Introduction
00:20 Motivation for Subspace Topology
09:51 Definition: Subspace Topology
11:04 Example: Union of Intervals
14:23 Example: Sequence 1/n
17:04 Prop: Open (closed) sets of containing space are open (closed) in subspace
19:07 Prop: If subspace is open (closed), then its open (closed) subsets are open (closed) in containing space
21:10 Characteristic Property of Subspace Topology
26:46 Extending and restricting continuous maps to subspaces
31:58 Prop: Properties of Subspaces

This lecture follows Lee's "Introduction to topological manifolds", chapter 3.

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