Math and science::Topology

# Metric space. Open and closed sets

### Open and closed subsets

Let $$X$$ be a metric space.

Open in $$X$$
A subset $$U$$ of $$X$$ is open in $$X$$ iff [...].
Closed in $$X$$
A subset $$V$$ of $$X$$ is closed in $$X$$ iff [...].

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.