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