The set $S = \{(x_1, x_2) \in \mathbb{R}^2 \mid x_1 + x_2 = 1\}$ can be analyzed as follows:

Step 1: Understanding the Set

• The set $S$ consists of all points in the plane $\mathbb{R}^2$ where the sum of the coordinates $x_1$ and $x_2$ equals 1.
• This is the equation of a straight line in the plane.

Step 2: Open Set

A set in $\mathbb{R}^2$ is open if, for every point in the set, there exists an open ball around that point that is entirely contained within the set.

• For any point $(x_1, x_2)$ on the line $x_1 + x_2 = 1$, any open ball centered at that point will include points where $x_1 + x_2 \neq 1$, which are not in $S$. Therefore, $S$ cannot contain any open ball around any of its points. Thus, $S$ is not open.

Step 3: Closed Set

A set is closed if it contains all its limit points, or equivalently, if its complement is open.

• The line $x_1 + x_2 = 1$ does include all of its limit points because any sequence of points on this line that converges will converge to a point still on the line. There are no "boundary" points outside the line that would belong to the closure of $S$ without being in $S$. Hence, $S$ is closed.

Conclusion

The set $S = \{(x_1, x_2) \in \mathbb{R}^2 \mid x_1 + x_2 = 1\}$ is closed but not open.

Would you like further details on the concepts discussed or have any questions?

Here are five related questions you might explore:

1. How do open and closed sets behave differently in $\mathbb{R}^n$ compared to $\mathbb{R}^2$?
2. What happens if we consider the set $S = \{(x_1, x_2) \mid x_1 + x_2 \leq 1\}$? Is it open, closed, or neither?
3. How do the properties of a set change if we define it with strict inequalities, like $x_1 + x_2 < 1$?
4. What is the significance of open and closed sets in defining continuity and limits?
5. How does the concept of a closed set relate to compactness in topology?

Tip: Visualizing the set on a coordinate plane can help in understanding whether it is open or closed. In this case, the line equation $x_1 + x_2 = 1$ clearly defines a boundary without an interior, aiding in determining that it is closed.