TLDR:
(1) what condition defines the class of sets $S$ such that for all $s\in S$ there exists topologies $T_1$ and $T_2$ on $s$ such that $T_1\cup T_2$ is not a topology on $s$?
(2) what condition defines the class of sets $U$ such that for all $u\in U$ the union of any [possibly finite] number of topologies on $u$ is a topology?
(3) for any set $X$, what condition defines the class/set of topologies $T$ on $X$ such that the union of any [possibly finite] number of members of $T$ is a topology on $X$?
Let $\mathbf{set}$ be the class of sets. For all $X\in\mathbf{set}$, let $\mathbf{top}(X)$ be the class/set of all topologies on $X$.
For all $X\in\mathbf{set}$, and topologies $T_1,T_2\in\mathbf{top}(X)$, we have that $T_1\cap T_2\in\mathbf{top}(X)$. However, it is not necessarily the case that $T_1\cup T_2\in\mathbf{top}(X)$.
(Paraphrased from my notes)
Not necessarily, i.e. it may be the case that the union of two topologies is a topology, or it may not. This begs the question when is the union of two topologies a topology, and when isn't it?
Here's my reasoning:
Suppose that every class of topologies contains at least two topologies such that their union is not a topology.
Proposition 1 (false):
$$\forall X\in \mathbf{set}.\exists T_1,T_2\in\mathbf{top}(X):T_1\cup T_2\notin\mathbf{top}(X)$$
Counterexample:
Let $X=\emptyset$.
$\mathbf{top}(X)=\{\{\emptyset\}\}$
$\forall T_1,T_2\in\mathbf{top}(X).T_1=T_2=\{\emptyset\}$
$\therefore \forall T_1,T_2\in\mathbf{top}(X).T_1\cup T_2\in \mathbf{top}(X)\quad\square$
Okay, so maybe it's just the empty set for which this is the case. Not a problem.
Proposition 2 (false):
$$\forall X\in\mathbf{set}\setminus\emptyset.\exists T_1, T_2\in\mathbf{top}(X):T_1\cup T_2\notin\mathbf{top}(X)$$
Counterexample:
Let $X=\{0,1\}$.
$\mathbf{top}(X)=\{\{\emptyset,X\},\{\emptyset,\{0\},X\},\{\emptyset,\{1\},X\},\{\emptyset,\{0\},\{1\},X\}\}$
$\forall T_1,T_2\in \mathbf{top}(X).T_1\cup T_2\in\mathbf{top}(X)\quad\square$
(This also covers the case of singleton sets, which I forgot to mention)
So... if not every set has topologies such that their union isn't a topology, then there must be a collection of sets $S$ such that for all $s\in S$ the union of any two topologies in $\mathbf{top}(s)$ is a topology. I would suppose that there is likewise a class of sets $P$ such that for all $p\in P$ the union of at least two topologies $\mathbf{top}(p)$ is not a topology. The question is: how do I define these classes?
(It is unclear at present whether the collections of such sets constitutes a set or a proper class. Please excuse the inconsistent use of 'set', 'class', and similar set-theoretic terms. I will correct this as soon as I am able.)
Specifically, I would like to show that each of the following is true, and define the class of all sets for which each holds:
(1) There exists a class of sets $S$ such that for all sets $X\in S$, there exist topologies $T_1$ and $T_2$, on $X$, such that $T_1\cup T_2$ is not a topology on $X$.
$$\exists S\subset\mathbf{set}:\forall X\in S.\exists T_1,T_2\in\mathbf{top}(X):T_1\cup T_2\notin\mathbf{top}(X)$$
(2) There exists a class of sets $U$, such that for all sets $X\in U$, the union of any number of topologies on $X$ is a topology on $X$. $$\exists U\subset\mathbf{set}:\forall X\in U.\forall T\in\mathcal{P}(\mathbf{top}(X))\setminus\emptyset.\bigcup_{\tau\in T}\tau\in\mathbf{top}(X)$$
(3) For every set $X$, there is a collection of collections of topologies $V$, on $X$, such that for every element $T\in V$, the union of all members of $T$ is a topology on $X$. $$\forall X\in\mathbf{set}.\exists V\subseteq\mathcal{P}^2(\mathbf{top}(X)):\forall T\in V.\bigcup_{T\in V} T\in\mathbf{top}(X)$$
(1) and (2) can easily be proven via example ((1) using your choice of infinite set and (2) using the examples of the emptyset and/or $\{0,1\}$), but I am unsure about (3).
Edit: Following Henno Brandsma's answer the class of sets specified in (1) is the class of all sets whose cardinality is $>2$, and the class of sets $U$ specified in (2) is its complement, the class of sets whose cardinality is $\leq2$.
Bonus question:
Is there a class of sets $W$ such that for all $X\in W$ the union of any [possibly finite] number of [nontrivial] topologies on $W$ is not a topology on $W$? My gut says 'no', but I'm not sure how to proceed with this one.
