Exercise on the Interior and Closure of a Set in Topology 2

In this exercise, we need to determine the interior \(\text{Int}(A)\) and the closure \(\text{Cl}(A)\) of the set \(A = \{a, c\}\) in the space \(X = \{a, b, c\}\) with the topology \(\{X, \emptyset, \{a\}, \{a, b\}\}\).

Open Sets in the Topology

The topology on \(X\) is \(\{X, \emptyset, \{a\}, \{a, b\}\}\). So, the open sets are:

\(X = \{a, b, c\}\)
\(\emptyset\)
\(\{a\}\)
\(\{a, b\}\)

Interior of \(A\) (\(\text{Int}(A)\))

The interior of \(A\) is the union of all the open sets contained within \(A\).

The open sets contained in \(A = \{a, c\}\) must be open subsets of \(A\).

The open sets we can consider are \(\emptyset\) and \(\{a\}\), since \(\{a, b\}\) is not contained in \(A\). Therefore:

$$ \text{Int}(A) = \{a\} \cup \emptyset = \{a\} $$

The interior of \(A = \{a\}\) is the set \( \{a\} \) itself.

Closure of \(A\) (\(\text{Cl}(A)\))

The closure of \(A\) is the intersection of all the closed sets that contain \(A\).

A closed set is the complement of an open set. So, the closed sets are:

\(X^c = \emptyset\)
\(\emptyset^c = X = \{a, b, c\}\)
\(\{a\}^c = \{b, c\}\)
\(\{a, b\}^c = \{c\}\)

The only closed set that contains $ A=\{a, c \} $ is the entire set $ X= \{a, b, c \} $, since no other closed set includes both \(a\) and \(c\).

Therefore, the closure of the set \(A\) is the entire set \(X = \{a, b, c\}\).

\[ \text{Cl}(A) = \{a, b, c\} \]

And so on.