I'm having problems with Example 3.6 of Awodey's Category Theory Book: In Top, the coproduct of two spaces $X+Y$ is their disjoint union with the topology $ O(X+Y) \cong O(X) × O(Y) $.
But I don't understand why $ O(X+Y) \cong O(X) × O(Y) $ holds true.
Looking at the site below(see atatched), I think $ O(X+Y) \cong O(X) + O(Y) $ is correct.
https://ncatlab.org/nlab/show/coproduct
I appreciate it if someone here can help me.
The topology on the disjoint union is $$O(X+Y) = \{U \subset X + Y \mid U \cap X \in O(X), U \cap Y \in O(Y) \} .$$A bit more abstract, let $\mathfrak P(S)$ denotes the power set of $S$. We define functions $$\phi : \mathfrak P(X+Y) \to \mathfrak P(X) \times \mathfrak P(Y), \phi(C) = (C \cap X, C \cap Y) ,$$ $$\psi : \mathfrak P(X) \times \mathfrak P(Y) \to \mathfrak P(X+Y) , \phi(A,B) = A + B .$$ Then $\psi \circ \phi = id$ and $\phi \circ \psi = id$, i.e. $\phi$ and $\psi$ are bijections which are inverse to each other.
We have $$O(X+Y) = \phi^{-1}(O(X) \times O(Y)) .$$ In other words, $\phi$ restricts to a bijection $$\Phi : O(X+Y) \to O(X) \times O(Y) .$$ This is the meaning of $$ O(X+Y) \cong O(X) × O(Y) \tag{1} .$$ Note that we do not have $ O(X+Y) = O(X) × O(Y) $. Formula $(1)$ only says that there is a canonical bijection between the sets on the LHS and RHS.
It is not true that $ O(X+Y) \cong O(X) + O(Y)$. This would mean that the open subsets of $X+Y$ are either open subsets of $X$ are open subsets of $Y$. No set of the form $A + B$ with $A \ne \emptyset$ and $B \ne \emptyset$ would be open. In other words, $O(X) + O(Y)$ is not closed under union.