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 for all \( u \in U \), there exists an \( \varepsilon > 0 \) such that \( B(u, \varepsilon) \subseteq U \).
- Closed in \( X \)
- A subset \( V \) of \( X \) is closed in \( X \) iff \( X \setminus V \) is open in \( X \).

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.

### Open, closed, neither or both

Subsets of a metric space can be any combination of open and closed.

### Open subset, closed subset

There's no such thing as an open set that is not an open *subset*; \( U
\) is open *in* \( X \). The importance of this is that a set \( U \) may
be open in one space and not open in another. \( [0, 1) \) is not open in \(
\mathbb{R} \) but is open in \( [0, 2] \).

## Context

#### Source

Tom Leinster's Topology notesComic: