An example of open closed continuous image of $T_0$-space that is not $T_0$ Engelking in his "General Topology" states that $T_0$ separation axiom is not preserved under open closed continuous maps.
But I can't find any example of open closed continuous image of $T_0$-space that is not $T_0$. Or, equivalently, an example of quotient space of $T_0$-space by open closed equivalence relation that is not $T_0$.
 A: Let $f:X \to Y$ be the desired open and closed continuous surjection. Since $Y$ is not $T_0$, it contains two topologically indistinguishable points. If we restrict $f$ to such two-point subset of codomain, we still get a desired continuous surjection, so we may assume that $Y$ is two-point indiscrete space. Hence, it is enough to find a $T_0$ space and its decomposition into two sets $A$, $B$ such that neither $A$ nor $B$ contains a nonempty closed or open subset of $X$. In particular, $X$ cannot contain isolated point nor closed point. In particular, it cannot be $T_1$ and it cannot be finite. So let's take some easy example of infinite $T_0$ space that is not $T_1$ and let's try to find a good decomposition…
A: @user87690, many thanks for the helpful hint)
Maybe, one of the simplest examples is the following.
On the set $\mathbb Z$ of all integers with left order topology which is obviously $T_0$ consider equivalence relation with partition consisting of the set of odd and the set of even integers. This relation is open and closed, since saturation of any nonempty set is $\mathbb Z$. So there are only two saturated open sets $\varnothing$ and $\mathbb Z$. Hence, the corresponding quotient space is two point indiscrete and therefore it is not $T_0$.
