. If U ⊂ R
n
is open and C ⊂ U is compact, show that there is a compact set D with interior Do
such
that C ⊂ Do and D ⊂ U.

To prove that if \( U \subset \mathbb{R}^n \) is open and \( C \subset U \) is compact, then there exists a compact set \( D \) with interior \( D^\circ \) such that \( C \subset D^\circ \) and \( D \subset U \), follow these steps:

### Given:
- \( U \subset \mathbb{R}^n \) is an open set.
- \( C \subset U \) is a compact set.

### To Prove:
There exists a compact set \( D \) such that:
1. \( C \subset D^\circ \) (the interior of \( D \)),
2. \( D \subset U \).

### Proof:

1. **Compactness of \( C \)**:
   - Since \( C \) is compact and \( U \) is open, \( C \) is contained in \( U \).
   - Compactness implies that \( C \) is closed and bounded in \( \mathbb{R}^n \).

2. **Open Covering of \( C \)**:
   - Because \( U \) is open and contains \( C \), for each point \( x \in C \), there exists an open ball \( B(x, r_x) \) centered at \( x \) with radius \( r_x > 0 \) such that \( B(x, r_x) \subset U \).

   - The collection of these open balls \( \{B(x, r_x) \}_{x \in C} \) forms an open cover of \( C \).

3. **Finite Subcover**:
   - Since \( C \) is compact, there exists a finite subcover from the above collection. Let's denote these balls as \( B(x_1, r_1), B(x_2, r_2), \ldots, B(x_m, r_m) \).
   - Therefore, \( C \subset \bigcup_{i=1}^m B(x_i, r_i) \subset U \).

4. **Construction of \( D \)**:
   - Define \( D = \overline{B(x_1, r_1)} \cup \overline{B(x_2, r_2)} \cup \ldots \cup \overline{B(x_m, r_m)} \).
   - Each \( \overline{B(x_i, r_i)} \) is compact (as the closure of a bounded and closed set in \( \mathbb{R}^n \) is compact), and the finite union of compact sets is compact. Therefore, \( D \) is compact.

5. **Verification of Conditions**:
   - By construction, \( C \subset \bigcup_{i=1}^m B(x_i, r_i) \subset D^\circ \). Therefore, \( C \subset D^\circ \).
   - Also, \( D \subset \bigcup_{i=1}^m \overline{B(x_i, r_i)} \subset U \), since \( U \) is open and contains each of these balls.

### Conclusion:
We have constructed a compact set \( D \) such that \( C \subset D^\circ \) and \( D \subset U \). This completes the proof.

Would you like more details, or do you have any questions?

Here are some related questions:
1. What is the difference between an open set and a compact set in \( \mathbb{R}^n \)?
2. How does the Heine-Borel theorem apply to compact sets in \( \mathbb{R}^n \)?
3. Can you generalize this result to other topological spaces?
4. What happens if \( C \) is not compact but only bounded?
5. How does this proof relate to the concept of the closure of a set?

**Tip:** Understanding the relationship between open and compact sets in \( \mathbb{R}^n \) is crucial for various concepts in analysis, such as continuity and compactness.

Explore the relationship between open and compact sets in \( \mathbb{R}^n \) with a detailed proof. Learn how to show that if \( U \subset \mathbb{R}^n \) is open and \( C \subset U \) is compact, there exists a compact set \( D \) with interior \( D^\circ \) such that \( C \subset D^\circ \) and \( D \subset U \). Suitable for advanced undergraduate students studying real analysis.

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