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

# Closed set (of reals), definition

A subset $$E \subseteq \mathbb{R}$$ is said to be closed if $$\overline{E} = E$$.

In other words, $$E$$ contains all of its adherent points. The operation which is implicit is the limit; any limt constructed from elements of $$E$$ will equal an element in $$E$$.

### Example

$$[a, b], (-\infty, b]$$, $$[a, \infty) \text{ and } (-\infty, \infty)$$ are closed, while $$(a, b), [a, b), (a, b], (-\infty, b), (a, \infty )$$ are not.

$$\mathbb{N}, \mathbb{Z}, \mathbb{R} \text{ and } \emptyset$$ are closed, while $$\mathbb{Q}$$ is not.

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