Contents of Disjoint Union Topology If I understand correctly (a big assumption), then if I have two disjoint spaces $X$ and $Y$, and the topology of $X$ is $\{X,A,B,A \cup B, \oslash\}$ and the topology of $Y$ is $\{Y,P,Q,P \cup Q, \oslash\}$, then the topology of their disjoint union would be $\{X \cup Y,A,B,A \cup B,P,Q,P \cup Q, \oslash\}$. But, if any union of the open sets of a topology must be a member of the topology, why is, for example, $A \cup P$ not in the disjoint union topology?
 A: The topology of he disjoint union of $X$ and $Y$ is larger than the one that you wrote. In fact, since it consists of those subsets $M$ of $X\cup Y$ such that $M\cap X$ is an open subset of $X$ and $M\cap Y$ is an open subset of $Y$, it does contain $A\cup P$.
A: Assumed $A\cap B=\emptyset$ and $P\cap Q=\emptyset$, in order that the given sets indeed form a topology, the topology (=set of open sets) of the disjoint union will be
$$\{\emptyset,\ A,\ B,\ A\cup B,\ X,\\
 P,\ Q,\ P\cup Q,\ Y,\\ 
A\cup P,\ A\cup Q,\ A\cup P\cup Q,\ A\cup Y, \\
 B\cup P,\ B\cup Q,\ B\cup P\cup Q,\ B\cup Y, \\
 A\cup B\cup P,\ A\cup B\cup Q,\ A\cup B\cup P\cup Q,\ A\cup T, \\
X\cup P, \ X\cup Q, \ X\cup P\cup Q,\ X\cup Y\} $$
A: For any topological spaces $X$ and $Y$, denoting the topology of $X$ by $T_X$ and of $Y$ by $T_Y$, and denoting the disjoint union as $X \coprod Y$, the disjoint union topology $T_{X \coprod Y}$ is in one-to-one correspondence with the Cartesian product $T_X \times T_Y$, where the ordered pair $(U,V) \in T_X \times T_Y$ corresponds to $U \coprod V \in T_{X \coprod Y}$.
Your topology $T_X$ has $5$ elements, and $T_Y$ also $5$ elements, so $T_{X \coprod Y}$ should have $5 \times 5 = 25$ elements. You list only $8$ elements though, so yes, your list is missing $A \coprod P$ as well as $16$ other elements.
