Consider the subset of R² defined as follows: A={(x,y)∈RxŔ:(x-1)(x-2)(y-3)(y+4)=0}

Which of the following statements are true?

A is connected

A is compact

A is closed

A is dense

Let's analyze the set \( A = \{(x, y) \in \mathbb{R}^2 : (x - 1)(x - 2)(y - 3)(y + 4) = 0\} \).

### Step 1: Understanding the structure of \( A \)
The equation defining \( A \) is:
\[
(x - 1)(x - 2)(y - 3)(y + 4) = 0
\]
This means that \( A \) consists of points where at least one of the factors \( (x - 1) \), \( (x - 2) \), \( (y - 3) \), or \( (y + 4) \) is zero. We can break this into the following subsets:
- \( x = 1 \) for all \( y \in \mathbb{R} \) (this is a vertical line at \( x = 1 \)),
- \( x = 2 \) for all \( y \in \mathbb{R} \) (this is a vertical line at \( x = 2 \)),
- \( y = 3 \) for all \( x \in \mathbb{R} \) (this is a horizontal line at \( y = 3 \)),
- \( y = -4 \) for all \( x \in \mathbb{R} \) (this is a horizontal line at \( y = -4 \)).

Thus, \( A \) consists of four straight lines in \( \mathbb{R}^2 \):
- The vertical lines at \( x = 1 \) and \( x = 2 \),
- The horizontal lines at \( y = 3 \) and \( y = -4 \).

### Step 2: Analyzing the properties

#### 1. **Is \( A \) connected?**

A set is connected if it cannot be separated into two disjoint non-empty open sets. Here, \( A \) consists of four separate lines. These lines do not intersect, and each line can be considered a separate subset. Therefore, \( A \) can be separated into disjoint subsets, meaning that \( A \) is **not connected**.

#### 2. **Is \( A \) compact?**

A set is compact if it is both closed and bounded. While the set \( A \) is made of lines that extend infinitely in both directions, it is **unbounded**. Therefore, \( A \) is **not compact**.

#### 3. **Is \( A \) closed?**

A set is closed if it contains all of its limit points. The set \( A \) is defined by the equality \( (x - 1)(x - 2)(y - 3)(y + 4) = 0 \), which represents lines, and lines in \( \mathbb{R}^2 \) are closed sets. Hence, \( A \) is **closed**.

#### 4. **Is \( A \) dense?**

A set is dense in \( \mathbb{R}^2 \) if every point in \( \mathbb{R}^2 \) is either in the set or arbitrarily close to a point in the set. Since \( A \) consists of only four lines and the rest of \( \mathbb{R}^2 \) is not part of \( A \), there are many points in \( \mathbb{R}^2 \) that are not close to any points of \( A \) (e.g., points far from these lines). Therefore, \( A \) is **not dense** in \( \mathbb{R}^2 \).

### Conclusion
- \( A \) is **not connected**.
- \( A \) is **not compact**.
- \( A \) is **closed**.
- \( A \) is **not dense**.

### Would you like to explore these concepts further or dive deeper into any specific property?

Here are some related questions:
1. What are the definitions of connectedness and compactness in higher dimensions?
2. How do we prove if a set in \( \mathbb{R}^2 \) is closed?
3. Can a set in \( \mathbb{R}^2 \) be closed and not compact?
4. How can we find the closure of a set in \( \mathbb{R}^2 \)?
5. What are examples of dense subsets in \( \mathbb{R}^2 \)?

**Tip**: To determine whether a set is compact, always check if it is both closed and bounded!

Consider the subset of R² defined as follows: A = {(x, y) ∈ RxR: (x-1)(x-2)(y-3)(y+4)=0}. Which of the following statements are true? A is connected, A is compact, A is closed, A is dense.

Explore the properties of the subset A in R² defined by the equation (x-1)(x-2)(y-3)(y+4)=0. Understand if A is connected, compact, closed, and dense, with a detailed analysis of these topological concepts. Suitable for undergraduate-level students studying set theory and topology.

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