19  Open and closed sets in Euclidean space

Open balls define interior points, open sets, closed sets, boundaries, and closures in Euclidean space.

Exercises

Exercise 19.1 Show that any \(\epsilon\)-ball is open.

Exercise 19.2 Show that \(A \cup \partial A\) is closed.

Exercise 19.3 Show that if \(A \subset \mathbb{R}^n\) is closed and a sequence \(\mathbf{x}_i \in A\) converges to some \(\mathbf{x}\in \mathbb{R}^n\), then \(\mathbf{x}\in A\). This gives an equivalent characterization of closed subsets of \(\mathbb{R}^n\).

Exercise 19.4 Show that the closure of \(A\), the smallest closed set \(\bar{A}\) that contains \(A\), is equal to \(A\cup \partial A\).

Exercise 19.5 Show that \[A = \{ (x,0) \mid x \in [-1,1] \} \subset \mathbb{R}^2\] is closed.

Exercise 19.6 Show that \(\mathbb{Q} \subset \mathbb{R}\) is neither open nor closed. Show that the boundary of \(\mathbb{Q}\) is \(\mathbb{R}\). Show that the interior of \(\mathbb{Q}\) is empty. What is the closure of \(\mathbb{Q}\)?

Exercise 19.7 Let \[A = \{ (x,y) \in \mathbb{R}^2 \mid 0 \leq x < 1, \quad 0 \leq y < 1\}.\] Compute the interior of \(A\), the boundary of \(A\), the closure of \(A\).