The set in question is $\{(x_1, x_2) \in \mathbb{R}^2 \mid -1 < x_1 < 1, x_2 = 0\}$. Let's analyze whether this set is open, closed, neither, or both.

Analysis:

1. Description of the Set:

   • This set describes a line segment along the $x_1$-axis where $x_2 = 0$ and $x_1$ ranges between $-1$ and $1$, not including $-1$ and $1$ themselves.
   • The set can be represented as $\{(x_1, 0) \mid -1 < x_1 < 1\}$.

2. Openness:

   • A set is open if, for every point in the set, there exists an $\epsilon$-ball around that point that is entirely contained within the set.
   • In this case, the set is a line segment on the $x_1$-axis. Since the set is one-dimensional, there's no $\epsilon$-ball around any point in this set that stays entirely within the set when considering the entire $\mathbb{R}^2$ space. Thus, the set is not open in $\mathbb{R}^2$.

3. Closedness:

   • A set is closed if it contains all its boundary points (i.e., if its complement is open).
   • The boundary of this set in the $x_1$-axis consists of the points $x_1 = -1$ and $x_1 = 1$. However, these boundary points are not included in the set. Therefore, the set is not closed.

Conclusion:

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

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Would you like more details on this topic, or do you have any questions? Here are some related questions you might find interesting:

1. What are the definitions of open and closed sets in $\mathbb{R}^n$?
2. How can a set be both open and closed (clopen)?
3. Can you provide examples of sets that are neither open nor closed in $\mathbb{R}^2$?
4. How do we determine the