1. If f:X-->Y is a homeomorphism, the f(Cl(A)) = Cl(f(A)) for every $A \in X$

2. If f:X-->Y is a homeomorphism, the f( $\mathfrak{d}$(A)) = $\mathfrak{d}(f(A))$ for every $A \in X$

2. Originally Posted by r2dee6
1. If f:X-->Y is a homeomorphism, the f(Cl(A)) = Cl(f(A)) for every $A \in X$

2. If f:X-->Y is a homeomorphism, the f( $\mathfrak{d}$(A)) = $\mathfrak{d}(f(A))$ for every $A \in X$
Lemma 1. Let $f:X \rightarrow Y$ be a function on the indicated topological spaces. f is continuous iff for each subset A of X, $f(\overline{A} ) \subset \overline{f(A)}$.

(1) Since $f$ is continuous, $f(\overline{A} ) \subset \overline{f(A)}$by lemma 1. We know that $f^{-1}$ is also continuous. Thus, $f^{-1}(\overline{f({A})}) \subset \overline{f^{-1}f({A})}$ by lemma 1. We have $\overline{f(A)} \subset f(\overline{A})$. We conclude that $f(\overline{A} ) = \overline{f(A)}$.

(2) Since f is a homeomorphism, $f(\mathfrak{d}(A)) = f(\overline{A} \cap \overline{X \setminus A} ) = \overline{f(A)} \cap \overline{Y \setminus f(A)} = \mathfrak{d}(f(A))$.