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I am trying to understand some notes I am reading on line. Here is what I thought I read: Given that $X$ is a finite set with a $T_0$ topology then $f:X \to X$ is a homeomorphism if and only if $f$ is either one to one or onto.

Here is a $T_0$ topology on $X = \{a,b,c\}$: $\{\{\}, \{c\},\{a,c\},\{b,c\},\{a,b,c\}=X\}$. Here is a bijective function $f:X \to X$ by $f(a) = a$, $f(b)=c$ and $f(c)=b$.

The preimage of $\{c\}$ is $\{b\}$. But $\{b\}$ is not an open set in $X$ so $f$ is not a homeomorphism. ?

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    What you thought you read simply isn’t true. I suspect that what it actually said is that a *continuous* map $f:X\to X$ is a homeomorphism if and only if it is either one-to-one or onto. – 2017-01-04
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    I second Brian. Could you also give us a link to said notes? – 2017-01-05

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