# Closure, definition

To define a closure, we will utilize *ε-adherent points* and
*adherent points*. Sets of reals have adherent points analogous to
sequences of reals having limit points.

#### ε-adherent point

Let \( X \) be a subset of \( \mathbb{R} \), let \( \varepsilon > 0 \) be a
real and \( x \in \mathbb{R} \) be another real. We say that \( x \) is
*ε-adherent* to \( X \) iff there exists a \( y \in X \) which is ε-close
to \( x \) (i.e. \( |x - y| \leq \varepsilon \) ).

#### Adherent point

Let \( X \) be a subset of \( \mathbb{R} \), and let \( x \in \mathbb{R} \)
be a real. We say that \( x \) is an *adherent point* of \( X \) iff it
is ε-adherent to \( X \) for every \( \varepsilon > 0 \).

### Closure

Let \( X \) be a subset of \( \mathbb{R} \). The *closure* of \( X \),
sometimes denoted as \( \overline{X} \), is defined to be the set of all adherent
points of \( X \).

### Example

The number 1 is ε-adherent to the open interval (0, 1) for every \( \varepsilon > 0 \), and is thus an adherent point of the interval (0, 1). The number 2, in comparison is not 0.5-adherent to (0,1), so can't be an adherent point of the interval.

The closure of \( \mathbb{N} \) is \( \mathbb{N} \). The closure of \( \mathbb{Z} \) is \( \mathbb{Z} \). The closure of \( \mathbb{Q} \) is \( \mathbb{R} \). And the closure of \( \mathbb{R} \) is \( \mathbb{R} \). The closure of \( \emptyset \) is \( \emptyset \).

#### Closures of intervals

- Closure of \( (a, b), [a, b), (a, b], \text{ and } [a, b] \) is \( [a, b] \)
- Closure of \( (a, \infty) \) or \( [a, \infty) \) is \( [a, \infty) \)
- Closure of \( (-\infty, a) \) or \( (-\infty, a] \) is \( (-\infty, a] \)
- Closure of \( (-\infty, \infty) \) is \( (-\infty, \infty) \)