Confusion with open set and a subset of R2

[Heresy]

Homework Statement

Using the definition of an open set, prove that the subset $S = \left({a}_{1}, {b}_{1}\right)\times \left({a}_{2}, {b}_{2}\right)$ of the Euclidean space ${R}^{2}$ is open.

Homework Equations

The Attempt at a Solution

I so far don't have much of an attempt, as I am merely trying to figure out what the question is asking. If I interpret the (a1, b1) x (a2, b2) as two vectors on a plane then S would be a vector pointing outwards from the plane? Which doesn't seem like an open set in R2 to me.

I am not looking for a solution, but looking for a clarification of the question in mind.

Thanks!

 

[aPhilosopher]

(a₁, a₂) is an open interval on the real number line, just like in high school or calculus.

here, the X refers to the cartesian product. so (a₁, b₁) X (a₂, b₂) is the set that consists of tuples¹ (x₁, x₂) where x₁ is in (a₁, b₁) and x₂ is in (a₂, b₂). There is an unfortunate overlap of notation between vector/tuple notation and the notation for intervals.

In symbols and for general sets:

$$S_{1} \times S_{2} = \left{\{ (a, b) : a \in S_{1}, b \in S_{2} \right}\}$$1 - tuples are essentially vectors but with no necessarily assumed linear structure

 

[Heresy]

Thank you very much!