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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 [...].

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 [...].

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 [...].

Closure

Let \( X \) be a subset of \( \mathbb{R} \). The closure of \( X \), sometimes denoted as \( \overline{X} \), is defined to be [...].