Error in Book? Is it the Box Topology?

[Kreizhn]

Hey,

I'm reading through a book and have come across something that seems like an obvious error to me. The books says

If $(X,T_X)$ and $(Y,T_Y)$ are topological spaces, there's a standard way to define a topology on the Cartesian product $X \times Y$. If we let
$$\mathbb B = \{ O_X \times O_Y : O_X \in T_X, O_Y \in T_Y \}$$
then the topology generated by this basis is called the product topology on $X \times Y$

Now it's been a long time since I've done any topology, but isn't this the box topology rather than the product topology? I just want to make sure I'm not going crazy.

 

[owlpride]

I think the product topology and box topology coincide for finite products. If you are looking at the space X x Y in particular, then the product topology is generated by the set

$$\{ O_X \times Y :O_X \in T_X \} \cup \{X \times O_Y:O_Y \in T_Y \}$$

Finite intersections of these sets are open as well. In particular,

$$(O_X \times Y) \cap (X \times O_Y) = O_X \times O_Y$$

is open.

 

[Kreizhn]

Thanks.

I think you're right. Planetmath says that they coincide when the index set is finite.