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

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.

Explore the concept of open and closed sets in \( \mathbb{R}^2 \) through the example of the set \( S = \{(x_1, x_2) \in \mathbb{R}^2 \mid x_1 + x_2 = 1\} \). Learn why this set is closed but not open, with detailed explanations and related questions.

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