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 \( (-\infty, \infty) \)