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 $T_0$.

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    $\begingroup$ Perhaps you could share your thoughts about approaching the problem, even if in an unsuccessful way. Merely stating "I need X" might strike Readers as imperious. $\endgroup$ – hardmath Aug 27 '15 at 18:20
  • $\begingroup$ Also, it would help if you included your definition of $T_0$ (or at least provided a link). $\endgroup$ – Omnomnomnom Aug 27 '15 at 18:31
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    $\begingroup$ @Omnomnomnom: There is only one definition of $T_0$: if $x$ and $y$ are distinct points, then at least one of them has a nbhd that does not contain the other. $\endgroup$ – Brian M. Scott Aug 27 '15 at 18:35
  • $\begingroup$ @Brian ah... I find the $T_n$ notation confusing. Thanks. $\endgroup$ – Omnomnomnom Aug 27 '15 at 18:38

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…

  • $\begingroup$ Do you have reason to believe this is possible? $\endgroup$ – Cameron Buie Aug 27 '15 at 19:23
  • $\begingroup$ @CameronBuie: I tried to follow the logic of what is necessary, so if it is not possible, then it is not possible to satisfy the OP's requirement. And I think I have an example. What is “the easiest” example of infinite $T_0$ space that is not $T_1$? (I would have written a complete solution, I just thought that my answer could start as a hint.) $\endgroup$ – user87690 Aug 27 '15 at 19:28
  • $\begingroup$ I'm unaware of any $T_0,$ non-$T_1$ spaces with neither closed points nor open points, so I'm at a loss. $\endgroup$ – Cameron Buie Aug 27 '15 at 19:35
  • $\begingroup$ @CameronBuie: An example of a $T_0$ space that is not $T_1$ that we had in our lecture was $ω$ with final segments as open sets… $\endgroup$ – user87690 Aug 27 '15 at 19:40
  • $\begingroup$ Under that topology, $0$ is a closed point. $\endgroup$ – Cameron Buie Aug 27 '15 at 19:45

@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 $T_0$.

  • $\begingroup$ Yes, that's exactly the example I've had in my mind. :-) $\endgroup$ – user87690 Aug 28 '15 at 10:23

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