Understanding the difference between basis, topology, and open sets in finite products

Question

I understand that an open set is just a subset of a space that obeys the definition of a topology.

I understand that a basis is a collection of open subsets of a space such that every open set of the space is a union of members of the basis.

But I'm confused for product spaces:

Product spaces $(\prod X_i, T')$ are defined with a basis for $T'$ being generated by $\prod O_i$. So by the definition of a basis, each $\prod O_i$ should be an open set in $(\prod X_i, T')$. But to use a specific case: If $(0,1)\times(0,1)$ and $(2,3)\times(2,3)$ are open sets then their union should be an open set in $R^2$. But this defies the definition of a topology since no $O_1\times O_2$ equals this for $O_1, O_2$ being open subsets in $R$, respectively.

Can someone explain what I'm misunderstanding here?

Answer 1

The product topology on $X \times Y$ is specified by the base $$\mathcal{B} = \{O_1 \times O_2: O_1 \subset X \text{ open } ,O_2 \subset Y \text{ open }\}$$

and indeed all open sets of $X \times Y$ are by definition the unions of those basic open sets.

So $(0,1) \times (0,1) \cup (2,3) \times (2,3)$ is indeed open as a union of basic open sets. It's not in $\mathcal{B}$, but a base is not closed under unions, why would it be? Most bases in practice are not closed under unions. You seem to think that $\mathcal{B}$ are all open sets, but there are way many more (e.g. the open circle in the plane is also a union of open squares).

Answer 2

Clarification: Most general, the product topology is generated by the sub-base of $\pi_i^{-1}(\theta_i)$, where $\theta_i$ is a open set of the $i^{th}$ space and $\pi_i$ the canonical projection in the respective space. A sub-base is a family of open sets s.t. any set in base is a finite intersection of sets in the sub-base. So the sets in base are finite intersection of pre-image of canonical projection; it's hard to imagine but it's like fixing finite coordinates in a open set and leaving the rest free in the respective component of the product.

Said, that, a open set of the product is a union (of any cardinal) of this pseudo-boxes from the base. The base is like lego pieces and topology it's like all you can build with the base.

Extra-information: The point of having sub-bases is that, with them, you can build a topology from almost nothing. I mean, what you get is the smallest topology that contains the sub-base, which don't need to have any property. That's hoy we can give a topology to a so ware space like product (any cardinal)