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Math and science::Analysis::Tao::09. Continuous functions on R

# 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.

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. $$|x - y| \leq \varepsilon$$ ).

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

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