Exercise 4.2

Question

Exercise 2:
    If $f$ is a continuous mapping of a metric space $X$ into a metric space $Y$, prove that $$f(\overline{E})\subset\overline{f(E)}$$ for every set $E\subset X$. Show, by an
    example, that $f(\overline{E})$ can be a proper subset of $\overline{f(E)}$.

ghostofgarborg · Accepted Answer

$$ f^{-1}(\overline{f(E)}) = f^{-1} \left( \bigcap_{\substack{V \supseteq f(E) \ V \text{ closed}}} V \right) 
        = \bigcap_{\substack{V \supseteq f(E) \ V \text{ closed}}} f^{-1}(V) \supseteq \bigcap_{\substack{W \supseteq E \ W \text{ closed}}} W = \bar E $$
    Consequently, $f(\bar E) \subseteq \overline{f(E)}$. To see that the inclusion can be proper,
    let $X = (0,1)$ seen as a subspace of $\mathbb R$, and $f$ the inclusion into $\mathbb R$. 
    If we let $E = X$, then 
    $$ f(\bar E) = f((0,1)) = (0,1) \subsetneq \overline{(0,1)} = [0,1] $$

Exercise 4.2

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Exercise 4.2

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