Metric space. Open and closed sets

Tom Leinster describes the openness of $U$:

Thus, $U$ is open if every point of $U$ has some elbow room—it can move a little bit in each direction without leaving $U$.

Personally, I like the phrase: every element of an open set has a neighbourhood.

ε-balls

Open ε-balls are open, and closed ε-balls are closed. Consider trying to prove this. They are open on account of the definition of openness, not by their own definition alone, despite their names being suggestive.