The set $S = \{(x_1, x_2) \in \mathbb{R}^2 \mid -1 < x_1 < 1, x_2 = 0\}$ can be analyzed as follows:

Step 1: Understanding the Set

• The set $S$ consists of all points in the plane $\mathbb{R}^2$ where:
  • $x_2 = 0$ (this implies the points lie on the x-axis),
  • $x_1$ is constrained to the open interval $(-1, 1)$.

Thus, the set $S$ is a segment of the x-axis between -1 and 1, but it does not include the endpoints $(-1, 0)$ and $(1, 0)$.

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.

• In this case, for any point $(x_1, 0)$ in $S$, the points around it with non-zero $x_2$ (which would be required to form an open ball) would not satisfy $x_2 = 0$. Therefore, $S$ does not contain an 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 endpoints $(-1, 0)$ and $(1, 0)$ are limit points of $S$, but they are not included in $S$. Thus, $S$ does not contain all its limit points, so $S$ is not closed.

Conclusion

The set $S = \{(x_1, x_2) \in \mathbb{R}^2 \mid -1 < x_1 < 1, x_2 = 0\}$ is neither open nor closed.

Would you like further details on open and closed sets or have any questions?

Here are five related questions you might explore:

1. What are examples of sets that are both open and closed (clopen sets)?
2. How would the nature of the set change if the inequality for $x_1$ was non-strict (e.g., $-1 \leq x_1 \leq 1$)?
3. How do we define open and closed sets in higher dimensions?
4. What is the significance of limit points in determining whether a set is closed?
5. How would you describe the set geometrically if the condition on $x_2$ was different?

Tip: When determining whether a set is open or closed, always consider the definition of open balls and limit points in the context of the set.