1. The terms "open" and "closed" can refer to different concepts depending on the context, such as intervals in mathematics or sets in topology. 2. In the context of intervals on the real number line: - An **open interval** $(a,b)$ includes all real numbers $x$ such that $a < x < b$, but does **not** include the endpoints $a$ and $b$. - A **closed interval** $[a,b]$ includes all real numbers $x$ such that $a \leq x \leq b$, including the endpoints $a$ and $b$. 3. For example, the open interval $(1,5)$ contains numbers like 2, 3.5, and 4.999 but not 1 or 5. The closed interval $[1,5]$ contains all numbers between 1 and 5 including 1 and 5 themselves. 4. In topology, a set is **open** if for every point in the set, there exists a small neighborhood around that point which is also entirely within the set. A set is **closed** if it contains all its limit points (or equivalently, its complement is open). 5. To summarize: - Open means endpoints or boundary points are not included. - Closed means endpoints or boundary points are included. This explanation covers the basic meaning of open and closed in common mathematical contexts.