Let $A$ be a nonempty subset of a topological space $X$. Which of the following statements is true?
1. If $A$ is connected, then its closure $\bar{A}$ is not necessarily connected
2. If $A$ is path connected, then its closure $\bar{A}$ is path connected
3. If $A$ is connected, then its interior is not necessarily connected
4. If $A$ is path connected, then its interior is connected