I was given the following problem :
A bijection $\ f$ : (X, $\tau_1$ ) $\to$ (X, $\tau_2$) is a quotient map iff $\tau_1 = \tau_2$
But I think the statement is not true.
For example , take X = $\Bbb R$, $\tau_1$ = Fixed point topology on $\Bbb R$ with respect to the point $\ 0$ and $\tau_2$ = Fixed point topology on $\Bbb R$ with respect to the point $\ 1$. Take $\ f$ : (X, $\tau_1$ ) $\to$ (X, $\tau_2$) $\,$ as $\ f(0)=1$ ,$\,$$\ f(1)=0$ $\,$ and $\;$ $f(x)=x$ $\;$ $\forall x \neq 0,1$
In the above example f is a bijection and a quotient map but $\tau_1 \neq \tau_2$. Rather $\tau_1$ & $ \tau_2$ are homeomorphic to each other. So I think the problem should be - A bijection $\ f$ : (X, $\tau_1$ ) $\to$ (X, $\tau_2$) is a quotient map iff $\tau_1$ & $\tau_2$ are homeomorphic to each other.
Am I right? Please give me some suggestions. Thank you in advance.
You are quite right that the statement is false. However, it is not sufficient that $\tau_1$ and $\tau_2$ are homeomorphic, you also need $f(\tau_1) = \tau_2$, i.e. f is a homeomorphism. (E.g. using your example spaces but with the identity map, the spaces are homeomorphic but the map is not a quotient.)