Question

Let \( X \) be a topological space and \( A \subseteq X \). Given a subset \( S \) of \( X \), let \( {int}(S), \partial S, \) and \( \overline{S} \) denote the interior, boundary, and closure, respectively, of the set \( S \). Which one of the following is NOT necessarily true?

• A. \( {int}(X \setminus A) \subseteq X \setminus \overline{A} \)
• B. \( A \subseteq \overline{A} \)
• C. \( \partial A \subseteq \partial ({int}(A)) \)
• D. \( \partial (\overline{A}) \subseteq \partial A \)

Hint:

For topology problems, carefully analyze the definitions of set operations like interior, closure, and boundary to verify logical inclusions.

Answer 1

The Correct Option is C

Option (1): The interior of \( X \setminus A \), denoted \( {int}(X \setminus A) \), consists of all points in \( X \setminus A \) that are not limit points of \( A \). This is indeed a subset of \( X \setminus \overline{A} \) because \( \overline{A} \) includes all points of \( A \) as well as its limit points. Hence, this is true. Option (2): By definition of the closure \( \overline{A} \), we have \( A \subseteq \overline{A} \), as the closure of \( A \) includes all points of \( A \) and its limit points. Hence, this is true. Option (3): The boundary of \( A \), \( \partial A \), consists of points that are in \( \overline{A} \setminus {int}(A) \). However, \( \partial ({int}(A)) \) is the boundary of the interior of \( A \), which may not always include all boundary points of \( A \) (e.g., points that are in the closure of \( A \) but not in \( {int}(A) \)). Hence, this is not necessarily true. Option (4): The boundary of \( \overline{A} \), \( \partial (\overline{A}) \), is a subset of \( \partial A \) because the closure of \( A \) does not introduce any additional boundary points. Hence, this is true. Final Answer: (3).