Is the next map a quotient map?

Question

Let $X=[0,1] cup (2,3]$, let $Y=[0,2]$ and suppose $X$ and $Y$ have the usual topologies. Define $f: X to Y$ by $f(x)=x$ if $x in [0,1]$ and $f(x)=x-1$ if $x in (2,3]$.
I tried proving that the function was closed, but I had problems. Can anyone give me a suggestion?

Brian M. Scott · Answer

HINT: Is $(1,2]$ closed in $Y$? Is $f^{-1}big[(1,2]big]$ closed in $X$? What would $f$ being a quotient map say about these two sets?

Is the next map a quotient map?

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