In Chapter 5 we study the topology of the real numbers, but one can also study the topology of general sets, where the open, closed and compact properties likewise generalize. Topology also classifies shapes, and at times considers shapes the same if one can one continuously deform one into the other (cue a joke about coffee mugs and doughnuts). Other shapes like Möbius strips are notable for their unique properties. Here's a video where this topology comes up naturally.
Want another 10 minute introduction to the broader field of topology? Check out this video from PBS Infinite Series.
Find someone who looks at you the way this guy looks at a Klein bottle. That is all.
Example B.2 of Appendix B discusses a theorem which implies that there are two antipodal points on the Earth which have exactly the same temperature. This video from 3Blue1Brown discusses other connections to this result, including a discrete version in which a pair of thieves are trying to evenly split their stolen necklace.
Example B.7 of Appendix B deals with how to find stable footing for a good table situated on uneven ground. This video also addresses this problem, plus adds an additional application to so-called hugging squares. They are two fun applications of the intermediate value theorem (Theorem 6.38).
Space-filling curves are continuous (two-dimensional) curves which somehow manage to completely fill up three-dimensional space. A famous such curve is called the Hilbert curve and is discussed in Example B.14 of Appendix B. A real-world application of this curve to videography and audiology is discussed in this video.