The exercises are not computational; they are theoretical. Many ask the student to prove, for example, that a finite topological space is compact, or that the continuous image of a connected set is connected. This is where solutions become invaluable.
Finally, we show that $\overlineA$ is the smallest closed set containing $A$. Let $B$ be a closed set such that $A \subseteq B$. We need to show that $\overlineA \subseteq B$. Let $x \in \overlineA$. Suppose that $x \notin B$. Then, there exists an open neighborhood $U$ of $x$ such that $U \cap B = \emptyset$. This implies that $U \cap A = \emptyset$, which contradicts the fact that $x \in \overlineA$. Therefore, $x \in B$, and hence $\overlineA \subseteq B$. Introduction To Topology Mendelson Solutions
Show that compact subset of Hausdorff space is closed. The exercises are not computational; they are theoretical