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 [...].

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