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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 \( E \).


\( [a, b], (-\infty, b] \), \( [a, \infty) \text{ and } (-\infty, \infty) \) 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.