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I was asked whether the following statement is true or false:

For a continuous function $f$, $f^{-1}(F)$ is closed whenever $F$ is closed.

I proved earlier that for open sets F this statement is true. After thinking for a while I came up with the following 'counterexample': $$ f: (0,1) \rightarrow [0,1] \\ f(x) = \begin{cases} 0 \hspace{3cm} x \in (0,1/4)\\ 2(x-1/4) \hspace{1.3cm} x \in [1/4,3/4]\\ 1 \hspace{3cm} x \in (3/4,1)\end{cases}$$

I know however that the statement should be true, as proven in https://math.stackexchange.com/a/107299/681496. Can anyone point me to the error in my counterexample?

Jeroen Nelis
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    In your example, what is the closed set $F\subseteq [0,1]$ whose preimage, you think, is not closed ? This function is continuous, so I assure you that you will not find such a set $F$, but let's hear from you. – Medo Aug 02 '21 at 13:49
  • I think that the pre-image of $F = [0,1]$ is equal to $(0,1)$ and thus not closed. Or is the pre-image here only $[1/4,3/4]$? – Jeroen Nelis Aug 02 '21 at 13:56
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    Recall that $(0,1)$ is closed in $(0,1)$. – Michael Hoppe Aug 02 '21 at 14:16

3 Answers3

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$A\subset X$ is closed $\iff X\setminus A$ is open. So if you know that preimages of open sets under a continuous map $f\colon X\to Y$ are open, you can do the following:

Let $B\subset Y$ be closed, i.e. $Y\setminus B$ is open. Since $f$ is continuous, $f^{-1}(Y\setminus B) = X\setminus f^{-1}(B)$ is open in $X$.

Thus $f^{-1}(B)$ is closed.

Zest
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Note that the pre image of $[\frac{1}{4},\frac{3}{4}]$ is $[\frac{3}{8},\frac{5}{8}]$., which is closed

Ron Abramovich
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When we are giving a counter example we have to be very specific at which point our example is not following the rule given in statement. You have given an example of a continuous function. Now you want to say that the statement is not true that means you should have found at least one closed set whose inverse image is not closed. But in your answer you have not given any closed set. So this little things should be taken care while giving counter example. Coming to the statement yes the statement is true as you have said before.

Shreya Chauhan
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