# Confusion with open set and a subset of R2

1. Sep 30, 2009

### Heresy

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. 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!

2. Sep 30, 2009

### aPhilosopher

(a1, a2) is an open interval on the real number line, just like in highschool 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

Last edited: Sep 30, 2009
3. Sep 30, 2009

### Heresy

Thank you very much!