Proposition

Let $X$ be any set whatsoever, and let $\mathscr{T} = \mathscr{P}(X)$ (the power set of $X$, which is the set of all subsets of $X$), so every subset of $X$ is open. This is called the discrete topology on $X$, and $(X, \mathscr{T})$ is called a discrete space.
Let $Y$ be any set, and let $\mathscr{T} = \{ Y, \emptyset \}$. This is called the trivial topology on $Y$.
Let $Z$ be the set $\{ 1, 2, 3 \}$, and declare the open subsets to be $\{ 1 \}, \{ 1, 2 \}, \{ 1, 2, 3 \}$, and the empty set.

Solution

$X$ and $\emptyset$ are open subsets.
Any intersection of finitely many open subsets of $X$ is a subset of $X$, so it is in $\mathscr{T}$.
Any union of arbitrarily many open subsets of $X$ is a subset of $X$, so it is in $\mathscr{T}$.

$Y$ and $\emptyset$ are open subsets.
The only sets in $\mathscr{T}$ are $\emptyset$ and $Y$. Any intersection of them is either $\emptyset$ or $Y$, so it is in $\mathscr{T}$.
Any union of elements in $Y$ is either $\emptyset$ or $Y$, so it is in $\mathscr{T}$.

$Z$ and $\emptyset$ are both open.
Since $\emptyset \subset \{ 1 \} \subset \{ 1, 2 \} \subset \{ 1, 2, 3 \}$, any intersection of them is just the “smallest” one. Thus an intersection of any elements in $Z$ is in $Z$.
Similarly, any union of them is just the “largest” one. Thus a union of any elements in $Z$ is in $Z$.