Intersection

The question is this: Show by example that the intersection of infinitely many open sets need not be open.

"The intersection of a finite number of open sets is always open," Georg explains, "but this is kind of an interesting problem. We have infinity under our fingers, isn't that kind of neat? We get to use an example that goes on forever."

Georg stares at the chalkboard for a little bit, and then he puts down his chalk and wipes his fingers on his sweater. At the table, Richard looks up from his phone. "No ideas?"

Georg shakes his head. "You?"

Richard puts down his phone and regards the board for a minute. They do their problem set on Wednesday nights in the small study room on the third floor of the library. It's the Western facing corner, and they always get there right after dinner - just in time for the sunset. It's kind of a centering experience.

Richard: "I've been thinking about signing up for the swim class."

Georg: "You don't know how to swim?"

Richard: "No, I do, but what if I just showed up and pretended I didn't know how to and then acted like I was the fastest learner ever? The swim girls teach that class to raise money for the team."

Georg looks back at the board. "Yeah, I guess."