\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \newcommand{\groupMul}[1] { \cdot_{\small{#1}}} \newcommand{\groupAdd}[1] { +_{\small{#1}}} \newcommand{\inv}[1] {#1^{-1} } \newcommand{\bm}[1] { \boldsymbol{#1} } \require{physics} \require{ams} \require{mathtools} \)

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

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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Source

p216