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$ f:X \rightarrow Y$ be a continuos map and $c \in Y$ Is $\{x \in X | f(x)=c\}$ necessarily a closed subset of $X$?


I was thinking about a contradiction. Can this be a counterexample? $f(x) = \frac{1}{x^2+1}$ is continuous on the whole real line and and the image of the set $[0,\infty)$ is the set $(0,1]$. So we have found a closed set, whose image under a continuos function is not closed(and nor open).

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The question isn’t about the images of subsets of $X$: it’s about the inverse images of singleton subsets of $Y$. –  Brian M. Scott Dec 18 '12 at 12:22
I believe it is not necessarily a closed subset. so what can be a decent counterexample? –  John Lennon Dec 18 '12 at 12:25
See my answer; it’s the simplest choice of $Y$ that works to give a counterexample (though not hte simplest choice of $X$). –  Brian M. Scott Dec 18 '12 at 12:29
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2 Answers

up vote 2 down vote accepted

If $Y$ is a $T_1$-space, so that singleton sets are closed, then $\{x\in X:f(x)=c\}$ is just the inverse image under $f$ of the closed set $\{c\}$ in $Y$, so by continuity of $f$ it is closed in $X$.

If there is a $c\in Y$ such that $\{c\}$ is not closed in $Y$, then $\{x\in X:f(x)=c\}$ need not be closed in $X$. For example, let $Y=\{0,1\}$, and let the open set of $Y$ be $\varnothing, \{1\}$, and $\{0,1\}$. ($Y$ is the Sierpiński space.) Note that $\{0\}$ is not open in $Y$, so $\{1\}$ is not closed. Define $f:\Bbb R\to Y$ by

$$f(x)=\begin{cases}0,&\text{if }x\le 0\\1,&\text{if }x>0\end{cases}\;;$$

it’s easy to check that $f$ is continuous, but $\{x\in\Bbb R:f(x)=1\}$ is not closed in $\Bbb R$.

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Thank you so much! –  John Lennon Dec 18 '12 at 13:27
@vercammen: You’re very welcome. –  Brian M. Scott Dec 18 '12 at 13:33
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Hint: You will not find a counterexample where $Y$ is the real line. This is because points are closed in the real line, and if $f$ is continuous then the inverse images of closed sets are closed. (Note that $\{ x \in X : f(x) = c \} = f^{-1} [ \{ c \} ]$.) This should lead you to consider topological spaces where points are not closed. It doesn't have to be complicated, $Y = \{ 0 , 1 \}$ with the indiscrete (trivial) topology will do. Now just fill in $X$ and an appropriate (continuous) function $f$.

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thanks for help! –  John Lennon Dec 18 '12 at 13:29
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