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Here is Prob. 3, Chap. 4 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:

Let $f$ be a continuous real function on a metric space $X$. Let $Z(f)$ (the zero set of $f$) be the set of all $p \in X$ at which $f(p)=0$. Prove that $Z(f)$ is closed.

I think I can give a proof of this result, even when $f$ is a continuous mapping of a metric space $X$ into a metric space $Y$ and when we replace $0$ by any particular point $q$ of $Y$. Since $\left\{q \right\}$ is a closed set in $Y$, so is its inverse image under any continuous mapping $f$, a fact which boils down to Rudin's result if we replace $Y$ by the usual metric space $\mathbb{R}$ and $q$ by the real number $0$. Am I right?

In fact, this result holds even when $X$ is any topological space, as long as $Y$ is a $T_1$ topological space. Am I right?

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  • $\begingroup$ It is true for every space $X$. The topological def'n of continuity of $f:X\to B$ is that $f^{-1}T$ is open in $X$ whenever $T$ is an open subset of $B.$ Equivalently that $f^{-1}T$ is closed in $X$ whenever $T$ is a closed subset of $B.$ There are also other equivalent def'ns. $\endgroup$ – DanielWainfleet Mar 10 '17 at 3:35
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Both remarks are correct, if $\{y\}$ is closed in $Y$ and $f:X \rightarrow Y$ is continuous then $f^{-1}[\{y\}]$ is closed.

And $Z(f) = f^{-1}[\{0\}]$, and $\mathbb{R}$ is $T_1$. Also $Z(f) = \cap_n f^{-1}[(-\frac{1}{n}, \frac{1}{n})]$ is a $G_\delta$ set, as every singleton in the reals is a closed $G_\delta$.

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