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Math Help - Question about homeomorphism

  1. #1
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    Question about homeomorphism

    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
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  2. #2
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    Quote Originally Posted by r2dee6 View Post
    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)).
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