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