What is Neighborhood, open set?

Example 12.2

Open intervals are open: for $x \in \intoo{a}{b}$, take $r = \min(x - a,\, b - x) > 0$. Half-lines $\intoo{a}{+\infty}$ are open; $\R$ and $\emptyset$ are open (the latter vacuously). $\intcc{0}{1}$ is not open: no interval around $0$ stays inside.

Example 12.4 (Certifying openness with explicit radii)

Is $U = \{x \in \R : x^2 > 2\}$ open? Yes, and the certificate can be written down: $U = \intoo{-\infty}{-\sqrt2} \cup \intoo{\sqrt2}{+\infty}$, a union of two open half-lines, open by Proposition 12.3. Alternatively, argue point by point: for $x \in U$ with $x > \sqrt 2$, take $r = x - \sqrt2 > 0$: every $y \in \intoo{x - r}{x + r}$ satisfies $y > \sqrt 2$, hence $y^2 > 2$; symmetrically on the left. Both styles matter — the structural one (build from known open sets by unions and finite intersections) scales better, the $\varepsilon$-style one works when no structure is visible; and Chapter 13 will add a third, the most powerful: $U$ is the preimage of the open $\intoo{2}{+\infty}$ under the continuous $x \mapsto x^2$.

Example 12.8

Segments $\intcc{a}{b}$, half-lines $\intco{a}{+\infty}$, finite sets, $\Z$ (a convergent sequence of integers is eventually constant) are closed. $\intoc{0}{1}$ is neither open (fails at $1$) nor closed ($\frac 1n \to 0 \notin$ the set): most sets are neither. $\R$ and $\emptyset$ are both open and closed — and they are the only such subsets of $\R$ (Exercise 12.9).