Principles of Mathematical Analysis, Rudin, 3th ed, Chapter 4, Problem 1

Problem

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. (\overline{E} denotes the closure of E.) Show, by an example, that f(\overline{E}) can be a proper subset of \overline{f(E)}.

Answer

For y \in f(\overline{E}), there exists x \in \overline{E} such that y=f(x). For \epsilon > 0, there exists \delta > 0 such that d_X(x,x')<\delta  \Rightarrow  d_Y(f(x),f(x'))<\epsilon. Since x \in \overline{E}, there exists z \in E such that d_X(x,z) < \delta, which implies f(z) \in N_\epsilon(f(x)). This means y=f(x) \in \overline{f(E)}.

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