Let's call a space $(X,\tau)$ *minimal $T\_0$* if it is $T\_0$ and for every topology $\sigma\subseteq \tau$ with $\sigma \neq \tau$ we have that $(X,\sigma)$ is not $T\_0$ any more.


We say $x\leq y$ in a $T\_0$-space $(X,\tau)$ if and only if every open neighborhood of $x$ is an open neighborhood of $y$. It's easy to verify that for $T\_0$-spaces, this relation is a partial order. We call it the partial order *associated* to $(X,\tau)$.


Question: if $(X,\tau)$ is minimal $T\_0$, is the associated order total?



Is the associated order of a minimal $T_0$ space always total?
This conjecture is correct. I claim that if the topological space is minimal $T\_{0}$, then the specialization ordering is total. We shall prove this result by contrapositive, so we shall show that if the specialization ordering is not linear, the the topology is not a minimal $T\_{0}$. The idea in this proof is similar to the proof that Joel David Hamkins gave for finite spaces. If the specialization ordering is not a linear ordering, then we want to extend the specialization ordering to a larger partial ordering and give that larger partial ordering a smaller $T\_{0}$ topology. However, instead of extending the partial ordering all the way to a linear ordering, we shall extend the partial ordering in a minimal way so that we can control the new partial ordering and new topology since Zorn's lemma does not give us much control over the new topology.


Suppose that $(X,\mathcal{T})$ is a $T\_{0}$-space and let $\leq$ denote the specialization ordering on $X$. Suppose that $\leq$ is not a total ordering. 


Since $X$ is not total ordering, there are two element $p,q\in X$ with $p\not\leq q$ and $q\not\leq p$. We shall now extend the partial ordering $\leq$ to a larger partial ordering $\preceq$ on $X$ so that $p\preceq q$ in the least way.


Define $\preceq$ to be the relation such that $x\preceq y$ if and only if $x\leq y$ or $x\leq p,q\leq y$. I claim that $\preceq$ is a partial ordering, and the proof that $\preceq$ is a partial ordering is a routine verification.


Reflexivity: The ordering $\leq$ is clearly reflexive. 


Transitivity: Now assume that $x\preceq y$ and $y\preceq z$. Then we shall prove that $\preceq$ is transitive in the four following cases:


Case 1. $x\leq y;y\leq z$: Clearly $x\leq z$, so $x\preceq z$.


Case 2. $x\leq y;y\leq p,q\leq z$: In this case, $x\leq p,q\leq z$, so $x\preceq z$


Case 3. $x\leq p,q\leq y;y\leq z$: We have $x\leq p,q\leq z$, so $x\preceq z$


Case 4. $x\leq p,q\leq y;y\leq p,q\leq z$: We have $x\leq p,q\leq z$, so $x\preceq z$.


Symmetry: Suppose that $x\preceq y,y\preceq x$. Then we shall prove $x=y$ in the following four cases:


Case 1. $x\leq y;y\leq x$: Clearly $x=y$.


Case 2. $x\leq y;y\leq p,q\leq x$: This case is impossible for it implies that $q\leq x\leq y\leq p$ hence $q\leq p$ a contradiction.


Case 3. $x\leq p,q\leq y;y\leq x$: This case also impossible since it implies that $q\leq y\leq x\leq p$ which is also a contradiction.


Case 4. $x\leq p,q\leq y;y\leq p,q\leq x$: This case is impossible as well since it implies that $p\leq x\leq q$, a contradiction.


We therefore conclude that $\preceq$ is a partial ordering. Now give $X$ the topology $\mathcal{S}$ generated by the subbasis $(\downarrow^{\preceq}x)^{c}$ where
$\downarrow^{\preceq}x=\{r\in X|r\preceq x\}$. Then $\preceq$ is the specialization ordering on $\mathcal{S}$. I now claim that $\mathcal{S}\subseteq\mathcal{T}$ and we shall prove this claim by showing that each downset $\downarrow^{\preceq}z$ is closed in $\mathcal{T}$ in two cases.


Case 1: $q\not\leq z$. Suppose that $q\not\leq z$. Then $x\preceq z$ if and only if $x\leq z$ or $x\leq p,q\leq z$. However, since $q\not\leq z$, we have $x\preceq z$ if and only if $x\leq z$. Therefore, $\downarrow^{\preceq}z=\downarrow^{\leq}z$, so $\downarrow^{\preceq}z$ is closed in $X$.


Case 2: $q\leq z$. Suppose that $q\leq z$. Then $x\preceq z$ if and only if $x\leq z$ or $x\leq p,q\leq z$. However, since $q\leq z$, we conclude that $x\preceq z$ if and only if $x\leq z$ or $x\leq p$. Therefore $\downarrow^{\preceq}z=\downarrow^{\leq}z\cup\downarrow^{\leq}p$, so $\downarrow^{\preceq}z$ is closed in $\mathcal{T}$.


We have now shown that each set $\downarrow^{\preceq}z$ is closed in $(X,\mathcal{T})$, so each set $(\downarrow^{\preceq}z)^{c}$ is open in $(X,\mathcal{T})$. Since $\mathcal{S}$ is generated by the subbasis consisting of the sets of the form $(\downarrow^{\preceq}z)^{c}$, we conclude that $\mathcal{S}\subseteq\mathcal{T}$. Furthermore, we have $\mathcal{S}\neq\mathcal{T}$ since these spaces have different specialization orderings. 


$\textbf{Remark}$ The minimal $T\_{0}$-spaces are in a one-to-one correspondence with the class of all totally ordered sets since every totally ordered set has a unique minimal $T\_{0}$-topology. In particular, if $(X,\leq)$ is a totally ordered set, then the unique minimal $T\_{0}$-topology on $X$ is generated by the basis consisting of sets of the form $\{x\in X|x>x\_{0}\}$. In particular, if $A\subseteq X$, then $A$ is open if and only if $A$ is an upwards closed set and either $A$ has no least element or $A$ has a least element $a$ and $a$ has an immediate predecessor.


