Proposition 5.5.3.label Let $X, Y$ be topological spaces, $A \subset X$, and $f: X \to Y$ be continuous, then $f(\ol{A}) \subset \ol{f(A)}$.
Proof. Since $f$ is continuous, $f^{-1}(\ol{f(A)})$ is closed and contains $A$.$\square$
Proposition 5.5.3.label Let $X, Y$ be topological spaces, $A \subset X$, and $f: X \to Y$ be continuous, then $f(\ol{A}) \subset \ol{f(A)}$.
Proof. Since $f$ is continuous, $f^{-1}(\ol{f(A)})$ is closed and contains $A$.$\square$