Surjective and continuous map between Hausdorff spaces

Question

Can we say that a surjective and continuous map $p:X\to Y$ is a quotient map iff both $X$ and $Y$ are Hausdorff? If not, could you give me an example? I am terrible at finding counterexamples.

Answer 1

In a bit more generality, suppose that $\mathcal{O}, \mathcal{O}^\prime$ are two Hausdorff topologies on a set $X$ such that the topology $\mathcal{O}^\prime$ is strictly finer than $\mathcal{O}$. Then the identity function $\mathrm{id}_X : X \to X$ is a continuous surjection from $\langle X , \mathcal{O}^\prime \rangle$ onto $\langle X , \mathcal{O} \rangle$, however it is not a quotient mapping between these spaces.

Answer 2

No, this is not true; consider $X=[0,2\pi)$, $Y=S^1$, and the surjective continuous map $p(t)=e^{it}$. One can see that $p$ is not a quotient map because the continuous function $f:X\to\mathbb{R}$ defined by $f(t)=t$ is constant on the fibers of $p$, but does not descend to a continuous function on $Y$.