(January 2010 #1a partial) Determine whether or not the sets \[\{(x,y)\in\mathbb R^2:x+y<3\}\] and \[\{f\in C([-1,1]):f(0)=0\}\] are open, closed, or compact, where \(C([-1,1])\) is considered with \(\lVert\cdot\rVert_\infty\).
Solution
1. \(A_1 = \{(x,y) \in \mathbb{R}^2 : x + y < 3\}\) is open, not closed, not compact.
1.1. Open: the map \((x,y) \mapsto x+y\) is continuous and \(A_1 = (x+y)^{-1}((-\infty, 3))\) is the preimage of the open set \((-\infty,3)\).
1.2. Not closed: the sequence \((3 - 1/n, 0) \in A_1\) converges to \((3,0) \notin A_1\).
1.3. Not compact: \(A_1\) is unbounded (e.g. \(\{(-m, 0) : m \ge 1\} \subseteq A_1\)), and compact subsets of \(\mathbb{R}^2\) are bounded.
2. \(A_2 = \{f \in C([-1,1]) : f(0) = 0\}\) is closed, not open, not compact.
2.1. Closed: the evaluation map \(E: C([-1,1]) \to \mathbb{R}\), \(E(f) = f(0)\), is continuous (indeed \(|E(f)| \le \lVert f\rVert_\infty\)), and \(A_2 = E^{-1}(\{0\})\) is the preimage of the closed set \(\{0\}\).
2.2. Not open: the zero function \(0 \in A_2\), but every ball \(B_\epsilon(0)\) contains the constant function \(g \equiv \epsilon/2\), for which \(g(0) = \epsilon/2 \ne 0\), so \(g \notin A_2\); no ball around \(0\) is contained in \(A_2\).
2.3. Not compact: \(A_2\) is unbounded — the functions \(f_m(x) = mx\) satisfy \(f_m(0) = 0\) and \(\lVert f_m\rVert_\infty = m \to \infty\).
3. Q.E.D.