Exercise 2.3.2 (Definition of continuity using closed sets)

Question

Show that a function $f: X \rightarrow Y$ is continuous if and only if $f^{-1}(E)$ is a closed subset of $X$ for every closed subset $E$ of $Y$.

Elu Thingol · Accepted Answer

\begin{proof} We verify both directions.
\begin{description}
\item[$\implies$] Suppose that $f$ is continuous. For any closed subset $E$ of $Y$, its complement $Y\setminus E$ is open and so is $f^{-1}(Y \setminus E) = X \setminus f^{-1}(E)$; thus, $f^{-1}(E)$ must be closed by definition.

\item[$\impliedby$] Conversely suppose that $f^{-1}(E)$ is a closed subset of $X$ for every closed subset $E$ of $Y$. For any open subset $V$ of $Y$, its complement $Y \setminus V$ is closed and so is $f^{-1}(Y \setminus V) = X \setminus f^{-1}(V)$; thus, $f^{-1}(V)$ must be open by definition.
\end{description}
\end{proof}

Exercise 2.3.2 (Definition of continuity using closed sets)

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Exercise 2.3.2 (Definition of continuity using closed sets)

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