# Trouble understanding open sets in product topology.

I have the following doubt. Consider a manifold $$(M,\tau)$$ and its product topology $$\tau^2$$.

How is then an open set $$U\in\tau^2$$ defined?

Is it $$U=\bigcup_{i\in I} U_i\times V_i$$, where $$U_i,V_i\in\tau$$ or $$U=\bigcup_{i,j\in I}U_i\times V_j$$?

It seems to me that it is the first one because in the second one

\begin{align} U=\bigcup_{i,j\in I}U_i\times V_j=\bigcup_{i\in I}(\bigcup_{j\in I}U_i\times V_j)=\bigcup_{i\in I}U_i\times(\bigcup_{j\in I}V_j)=(\bigcup_{i\in I}U_i)\times\text{Im }U=\text{Dom }U\times\text{Im }U \end{align}

And, so, for example, $$U=A\times B\cup C\times D$$ could not be an open set if $$A$$ and $$C$$, and $$B$$ and $$D$$ are disjoint open sets in $$\tau$$.

Am I right? Thanks.

• I didn't follow your reasoning with the equalities, but it should be the second one. Think for example of what open sets look like in the plane $\mathbb R \times \mathbb R$. They are unions of little open boxes $(a,b) \times (c,d)$. – D_S Aug 28 at 13:42
• Note that both sets are open. Every open set in $\tau^2$ has the first form, but not nessecarily the second. An example is $(0,1)\times (0,1) \cup (2,3) \times (2,3) \subset \mathbb{R}^2$. – G. Chiusole Aug 28 at 14:11

It should be the first. An open set is a union of products of open sets, so it would take the form $$\bigcup_{i \in I} U_i \times V_i$$. It is a union of single terms (the terms just happen to have two factors) so only one index is needed. The second option you present is too big."
You should think of an open set as $$\bigcup_i \mathcal{O_i}$$, where each $$\mathcal{O}_i$$ has the form $$\mathcal{O}_i = U_i \times V_i$$. This makes it clear that one index suffices.