Here’s a characterization of maximal non-discrete topologies.
Lemma. Let $\tau$ be a non-discrete topology on a set $X$, and let $N=\big\{x\in X:\{x\}\notin\tau\big\}\ne\varnothing$. Then $\tau$ is maximal non-discrete iff
- $\tau$ induces the discrete topology on each $A\in\wp(X)\setminus\tau$, and
- $A\supseteq N$ whenever $A\in\wp(X)\setminus\tau$.
Proof. Assume first that $\tau$ is maximal non-discrete. Let $A\in\wp(X)\setminus\tau$. If $\tau_A$ is the topology generated by the subbase $\tau\cup\{A\}$, it’s not hard to check that $$\tau_A=\big\{U\cup(A\cap V):U,V\in\tau\big\}\;,$$ so every subset of $X$ must be expressible in the form $U\cup(A\cap V)$ for some $U,V\in\tau$. In particular, for each $x\in X$ there must be $U_x,V_x\in\tau$ such that $\{x\}=U_x\cup(A\cap V_x)$. If $\{x\}\in\tau$ we may take $U_x=\{x\}$ and $V_x=\varnothing$; if, however, $\{x\}\notin\tau$, we must have $U_x=\varnothing$ and $A\cap V_x=\{x\}$. Thus, $\tau$ induces the discrete topology on $A$.
Suppose that $A\nsupseteq N$ for some $A\in\wp(X)\setminus\tau$, and fix $x\in N\setminus A$. Then $\tau_A$ is strictly finer than $\tau$, but $\{x\}\notin\tau_X$, contradicting the maximality of $\tau$.
Now suppose that $\tau$ satisfies (1) and (2), let $\tau'$ be a topology strictly finer than $\tau$, and let $U\in\tau'\setminus\tau$. By hypothesis $U\supseteq N$ and $\tau$ induces the discrete topology on $U$, so $\{x\}\in\tau'$ for each $x\in N$, and $\tau'$ is the discrete topology on $X$. Thus, $\tau$ is maximal non-discrete. $\dashv$
But now we have an easy
Theorem. Let $\tau$ be a topology on a set $X$. Then $\tau$ is maximal non-discrete iff $X$ has a unique non-isolated point.
Proof. In the notation of the lemma this just says that $\tau$ is maximal non-discrete iff $N$ is a singleton. If $N$ is a singleton, then (1) and (2) are clearly satisfied, so $\tau$ is maximal non-discrete. Suppose now that there are distinct $x,y\in N$; then $\{x\}\in\wp(X)\setminus\tau$, but $\{x\}\nsupseteq N$, so $\tau$ is not maximal non-discrete. $\dashv$
In Michael Greinecker’s example the unique non-isolated point is of course $y$, and its nbhd filter is the fixed filter generated by the set $\{x,y\}$. Among the nicer examples are $\omega+1$ (i.e., a simple sequence with its limit point) and $\{p\}\cup\omega$ for any $p\in\beta\omega\setminus\omega$.
(I could probably have done this more simply, but this is how I actually discovered the result, so I thought that I’d let it stand as is.)
Added: I realized somewhat belatedly that these spaces can be described even more precisely. Let $X$ be a set with more than one element, and fix $p\in X$. Let $Y=X\setminus\{p\}$, let $\mathscr{F}$ be any filter on $Y$, and let $\tau=\big\{\{p\}\cup F:F\in\mathscr{F}\big\}\cup\wp(Y)$. Then $\tau$ is a maximal non-discrete topology on $X$, and all such topologies are obtained in this way.
