Homework Help: 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!

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