Is the set open or closed or neither or both. {(x_1,x_2) In R^2 such that -1 < x_1 <1 ,x_2=0}

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

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