Why is x > 0 not an open interval?

[maverick280857]

Hi,

This might be a trivial question, but I'll ask it anyway.

$$X=\{x\in\mathbb{R}\,|\,x > 0\}$$

is an open set. This is not, however, an open interval.

Why is this not an open interval? Let a, b be two elements in X, such that a < b. Then, every real number c satisfying the order relation a < c < b is necessarily in (a, b) and hence in X. Hence X is an interval. Since X is equivalent to the interval (0, $\infty$), I assumed it is also an open interval. Does the definition of an open interval only apply to intervals with finite end points? I thought otherwise: http://en.wikipedia.org/wiki/Interval_(mathematics)#Infinite_endpoints.

Thanks in advance.

 

[Studiot]

I don't know the reference and thus the context, but are you sure the authors are not referring to a 'bounded interval' ?

Or are the authors distinguishing between the reals (which do not contain $$\pm \infty$$ ) and the extended reals which do?
This distinction is often ignored.

 

[g_edgar]

Look in that book for the definition of "open interval". Your set is an open set and an interval, but perhaps there is a technical definition of "open interval" used in that book, requiring an "open interval" to have the form (a,b) with b a real number.

 

[maverick280857]

Thanks for the replies. This is what the book says

Open interval: (a, b) = { $x \in \mathbb{R}$ | $a < x < b$ }
Closed interval: [a, b] = { $x \in \mathbb{R}$ | $a \leq x \leq b$ }

A closed interval contains its end points, while an open interval does not.

Definition: $X \subset \mathbb{R} \implies x \in (a, b) \subset X$ for some (a, b)

"..a subset X of R will be called open if every point inside it can be enclosed by an open interval (a, b) lying entirely inside it."

No mention has been made of whether a and b must necessarily be finite.

 

[Studiot]

and what is the book's definition of R ?

does it contain infinity (the extended reals)

or not (the reals)

 

[mistermath]

What are a and b? Most likely, the book wants a,b to be elements of R.

 

[maverick280857]

Nothing else is listed explicitly. I would assume that a and b are elements of R too. But no mention is made of whether a and b can be infinite.

 

[mistermath]

If it turns out out that a,b are supposed to be members of R, then it makes sense that (5, infinity) is not an open interval (because open sets are the ones of the form (a,b) where both a AND b are in R. (infinity is not in R).

 

[maverick280857]

(infinity is not in R).

Ah. Thanks.

 

[Rasalhague]

If it turns out out that a,b are supposed to be members of R, then it makes sense that (5, infinity) is not an open interval (because open sets are the ones of the form (a,b) where both a AND b are in R. (infinity is not in R).

I guess you meant to write "open intervals" instead of "open sets". (The example was said to be an open set but not an open interval.)

 

[mistermath]

I guess you meant to write "open intervals" instead of "open sets". (The example was said to be an open set but not an open interval.)

You read my mind, thank you for the correction.