Is this the correct definition for the standard topology in $\Bbb R^2$

Question

Is this the correct definition for the standard topology in $\Bbb R^2$;

$T_{st}=\{(x_1,y_1)\times(x_2,y_2)|x_i,y_i\in \Bbb R\}$

Answer 1

Let $\mathbb R$ be equipped with its usual topology.

For $i=1,2$ there are projection functions $\pi_i:\mathbb R^2\to\mathbb R$ prescribed by $\langle x_1,x_2\rangle\mapsto x_i$.

Then the product topology is the smallest topology such that the projection functions are continuous.

Demanding this comes to the same as demanding that the product topology is the smallest topology such that sets of the form $(a,b)\times\mathbb R$ and $\mathbb R\times(a,b)$ are open sets.

Also it comes to the same as demanding that the product topology is the smallest topology such that sets of the form $(a,b)\times(c,d)$ are open sets. This corresponds with the collection mentioned in your question and this collection serves as a subbase and also (more strongly) as a base of that topology.

Be aware though that this collection is not a topology itself. For that note that e.g. the union $(0,2)\times(0,2)\cup(1,3)\times(1,3)$ is not an element of the collection, so the collection is not closed under unions (hence is no topology).

Further it can be proved that this product topology is the same as the topology that is induced by the metric: $$d(\langle x,y\rangle,\langle x',y'\rangle)=||\langle x,y\rangle-\langle x',y'\rangle||$$

The topology described is commonly labeled as "standard topology" on $\mathbb R^2$.