Exercise 284Topology › Topology of ℝⁿL1Added 3 Nov 2026

Open sets in $\mathbb{R}^n$

#open sets#definition#ball

Statement

State the definition of an open set in $\mathbb{R}^n$ and give two examples.

Hint

$U$ is open iff every point of $U$ has a small ball contained in $U$ .

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Definition. $U\subseteq\mathbb{R}^n$ is open iff for all $x\in U$ there is $r>0$ with $B(x,r)\subseteq U$ .

Examples. $\mathbb{R}^n$ itself, open balls, $\{x:x_1>0\}$ (open half-space).

Counterexample. The closed ball $\overline B(a,r)$ is not open (its boundary points fail the condition).

Conclusion. Openness is defined pointwise via balls.