Why is x > 0 not an open interval?

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In summary, the book defines an open interval as a subset of R containing its end points. The book also says that an open interval contains its end points if it exists, which excludes the set (5, infinity) because it does not contain its end points.
  • #1
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Hi,

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

Lectures on Advanced Mathematical Methods for Physicists by Sunil Mukhi and N. Mukunda said:
[tex]X=\{x\in\mathbb{R}\,|\,x > 0\}[/tex]

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, [itex]\infty[/itex]), 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.
 
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  • #2
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 [tex] \pm \infty [/tex] ) and the extended reals which do?
This distinction is often ignored.
 
  • #3
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.
 
  • #4
Thanks for the replies. This is what the book says

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

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

Definition: [itex]X \subset \mathbb{R} \implies x \in (a, b) \subset X[/itex] 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.
 
  • #5
and what is the book's definition of R ?

does it contain infinity (the extended reals)

or not (the reals)
 
  • #6
What are a and b? Most likely, the book wants a,b to be elements of R.
 
  • #7
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.
 
  • #8
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).
 
  • #9
mistermath said:
(infinity is not in R).

Ah. Thanks.
 
  • #10
mistermath said:
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.)
 
  • #11
Rasalhague said:
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.
 

1. Why is x > 0 not an open interval?

An open interval is defined as a set of real numbers that includes all values between two endpoints, but does not include the endpoints themselves. Therefore, the interval (0, ∞) would be considered an open interval because it includes all values greater than 0, but does not include 0 itself. In contrast, the inequality x > 0 includes 0 as a possible value, making it a closed interval.

2. What is the difference between an open interval and a closed interval?

The main difference between an open interval and a closed interval lies in their endpoints. An open interval does not include its endpoints, while a closed interval includes both of its endpoints. This means that for an open interval (a, b), a and b are not part of the interval, but for a closed interval [a, b], both a and b are included in the interval.

3. Can x > 0 ever be considered an open interval?

No, x > 0 can never be considered an open interval because it includes 0 as a possible value. An open interval, by definition, does not include its endpoints, so any interval that includes 0 cannot be considered open.

4. Why do we use open intervals in mathematics?

Open intervals are commonly used in mathematics because they provide a more precise representation of a range of values. For example, if we want to represent all positive real numbers, we can use the open interval (0, ∞) instead of the closed interval [0, ∞). This allows us to exclude 0, which is not considered a positive number. Additionally, open intervals are useful for continuity, differentiation, and other mathematical concepts.

5. How do open intervals relate to functions and their domains?

Open intervals are often used to define the domain of a function. For example, if we have a function f(x) = 1/x, the domain of this function would be (0, ∞) because the function is not defined for x = 0. In this case, using an open interval helps us to accurately represent the domain of the function and avoid any confusion or errors in calculations.

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