1. **Problem Statement:** We have a sequence of closed intervals $F_i = [a_i, b_i]$ with $F_1 \subseteq F_2 \subseteq F_3 \subseteq \cdots$. We want to provide examples showing that the union $S_\infty = \bigcup_{i=1}^\infty F_i$ can be: (a) open but not closed, (b) closed but not open, (c) open and closed, (d) neither open nor closed. 2. **Key Concepts:** - A set is **closed** if it contains all its limit points. - A set is **open** if for every point in the set, there exists an epsilon neighborhood fully contained in the set. - Closed intervals $[a,b]$ are closed sets. - The union of closed sets is not necessarily closed. 3. **(a) Example where $S_\infty$ is open but not closed:** Consider $F_i = \left[\frac{1}{i}, 1 - \frac{1}{i}\right]$ for $i \geq 2$. - Each $F_i$ is closed. - The union is $S_\infty = \bigcup_{i=2}^\infty \left[\frac{1}{i}, 1 - \frac{1}{i}\right] = (0,1)$. Explanation: As $i \to \infty$, $\frac{1}{i} \to 0$ and $1 - \frac{1}{i} \to 1$, so the union covers all points between 0 and 1 but excludes 0 and 1. - $(0,1)$ is open but not closed. 4. **(b) Example where $S_\infty$ is closed but not open:** Consider $F_i = [-i, i]$ for $i=1,2,3,\ldots$. - Each $F_i$ is closed. - The union is $S_\infty = \bigcup_{i=1}^\infty [-i, i] = \mathbb{R}$. Explanation: The union covers the entire real line. - $\mathbb{R}$ is both open and closed, but to satisfy "closed but not open," consider instead $F_i = [0, 1 - \frac{1}{i}]$. - Then $S_\infty = [0,1)$ which is closed on the left but open on the right, so not closed. To get a closed but not open union, consider $F_i = [0,1]$ for all $i$. - Then $S_\infty = [0,1]$ which is closed but not open. 5. **(c) Example where $S_\infty$ is both open and closed (clopen):** Consider $F_i = \mathbb{R}$ for all $i$. - Then $S_\infty = \mathbb{R}$ which is both open and closed. Alternatively, in a discrete topology, any set can be clopen, but in standard real analysis, only $\emptyset$ and $\mathbb{R}$ are clopen. 6. **(d) Example where $S_\infty$ is neither open nor closed:** Consider $F_i = \left[0, 1 - \frac{1}{i}\right]$ for $i=1,2,3,\ldots$. - Each $F_i$ is closed. - The union is $S_\infty = [0,1)$. - $[0,1)$ is closed on the left but open on the right, so it is neither open nor closed. **Summary:** - (a) $F_i = \left[\frac{1}{i}, 1 - \frac{1}{i}\right]$, $S_\infty = (0,1)$ open not closed. - (b) $F_i = [0,1]$, $S_\infty = [0,1]$ closed not open. - (c) $F_i = \mathbb{R}$, $S_\infty = \mathbb{R}$ open and closed. - (d) $F_i = \left[0, 1 - \frac{1}{i}\right]$, $S_\infty = [0,1)$ neither open nor closed.