) Let X = (a, b, c, d, e, f), t = {p, X, (c), (c, d), (c, f). [c, d, f)). For A = (a, c, d, e) evaluate a. int(A) b. ext(A) c. b(A) d. A e. relative topology Ta on A. (6+14-20)

To address the problems, let's clarify the notation and definitions.

### Given Set and Topology:

- **Set \( X = \{a, b, c, d, e, f\} \)**
  
- **Topology \( \mathcal{T} = \{\emptyset, X, \{c\}, \{c, d\}, \{c, f\}, \{c, d, f\}\} \)**
  
- **Subset \( A = \{a, c, d, e\} \)**

### Definitions and Solutions:

#### a. Interior of \( A \), denoted \( \text{int}(A) \):

The **interior** of \( A \) in topology \( \mathcal{T} \) is the largest open set in \( \mathcal{T} \) that is contained in \( A \).

1. The open sets in \( \mathcal{T} \) are: \( \emptyset, \{c\}, \{c, d\}, \{c, f\}, \{c, d, f\}, X \).
2. The open sets contained in \( A = \{a, c, d, e\} \) are: \( \emptyset, \{c\}, \{c, d\} \).

Thus, the **largest open set** contained in \( A \) is \( \{c, d\} \).

\[
\text{int}(A) = \{c, d\}
\]

#### b. Exterior of \( A \), denoted \( \text{ext}(A) \):

The **exterior** of \( A \) is the interior of the complement of \( A \) in \( X \).

1. \( A^c = X \setminus A = \{b, f\} \)
2. To find \( \text{ext}(A) \), we determine the largest open set in \( \mathcal{T} \) contained in \( A^c = \{b, f\} \).

From the open sets, the only subset of \( A^c = \{b, f\} \) is \( \emptyset \).

\[
\text{ext}(A) = \emptyset
\]

#### c. Boundary of \( A \), denoted \( b(A) \):

The **boundary** of \( A \), \( b(A) \), is defined as the set difference between the closure of \( A \) and the interior of \( A \).

1. The **closure** of \( A \), \( \text{cl}(A) \), is the smallest closed set containing \( A \). However, \( \text{cl}(A) = X \) (since \( X \) is the whole space).
2. \( b(A) = \text{cl}(A) \setminus \text{int}(A) \)

Given \( \text{int}(A) = \{c, d\} \), \( \text{cl}(A) = X = \{a, b, c, d, e, f\} \).

\[
b(A) = X \setminus \{c, d\} = \{a, b, e, f\}
\]

#### d. \( A \) itself:

\( A = \{a, c, d, e\} \)

#### e. Relative topology \( \mathcal{T}_A \) on \( A \):

The **relative topology** on \( A \), \( \mathcal{T}_A \), consists of intersections of the open sets in \( \mathcal{T} \) with \( A \).

1. Open sets in \( \mathcal{T} \): \( \emptyset, X, \{c\}, \{c, d\}, \{c, f\}, \{c, d, f\} \).
2. Intersections with \( A = \{a, c, d, e\} \):
   - \( \emptyset \cap A = \emptyset \)
   - \( X \cap A = \{a, c, d, e\} \)
   - \( \{c\} \cap A = \{c\} \)
   - \( \{c, d\} \cap A = \{c, d\} \)
   - \( \{c, f\} \cap A = \{c\} \)
   - \( \{c, d, f\} \cap A = \{c, d\} \)

Thus, the relative topology \( \mathcal{T}_A \) on \( A \) is:

\[
\mathcal{T}_A = \{\emptyset, \{c\}, \{c, d\}, \{a, c, d, e\}\}
\]

### Summary of Results:

a. \( \text{int}(A) = \{c, d\} \)  
b. \( \text{ext}(A) = \emptyset \)  
c. \( b(A) = \{a, b, e, f\} \)  
d. \( A = \{a, c, d, e\} \)  
e. \( \mathcal{T}_A = \{\emptyset, \{c\}, \{c, d\}, \{a, c, d, e\}\} \)  

Would you like more details on any part of this solution, or do you have any other questions?

### Follow-up Questions:
1. What is the definition of a **closed set** in topology?
2. How do you determine the **closure** of a set in a given topology?
3. What is the difference between **interior** and **closure**?
4. How does a **relative topology** differ from the original topology?
5. What are the properties of the **boundary** of a set?

**Tip:** Remember, in topology, the concepts of open, closed, interior, exterior, and boundary are relative to the topology defined on the set!

Explore the concepts of interior, exterior, boundary, and relative topology in set theory and topology. Learn how these concepts are applied to sets and topologies, with detailed examples and solutions.

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