Exercise 4.4

Question

Exercise 4:
    Let $f$ and $g$ be continuous mappings of a metric space $X$ into a metric space $Y$, and let $E$ be a dense subset of $X$. Prove that $f(E)$ is dense in $f(X)$. If $g(p)=f(p)$
    for all $p\in E$, prove that $g(p)=f(p)$ for all $p\in X$.

ghostofgarborg · Accepted Answer

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Let $U \subset f(X)$ be open. Then $f^{-1}(U)$ is open, and since $E$ is dense, it contains points of $E$.
    Therefore $U$ contains points of $f(E)$. This makes $f(E)$ dense in $f(X)$.

Let $p \in X$, and $p_n$ a sequence in $E$ that converges to $p$. By theorem 4.6 and the 
    equality of $f$ and $g$ on $E$
    $$ f(p) = \lim_{n \to \infty} f(p_n) = \lim_{n \to \infty} g(p_n) = g(p) $$
    so that $f$ and $g$ agree on all of $X$.

Exercise 4.4

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Exercise 4.4

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