Does the Interval [-1,∞) Include Any Open Sets?

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In summary, the question is whether the half open interval [-1, ∞) on the real number line contains an open set. While it includes -1, it is not considered an open set. However, it does contain the open set (4, 7) as an example. The difference in the question lies in the definition and specification of open and closed sets.
  • #1
zli034
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On the number line R, does [-1,[tex]\infty[/tex]) contain an open set?

because it includes -1, don't think it is an open set.
 
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  • #2
zli034 said:
On the number line R, does [-1,[tex]\infty[/tex]) contain an open set?

because it includes -1, don't think it is an open set.

It's a half open interval that you've shown. If you define a set {[tex][-1,\infty)[/tex]} then it contains at least one half open subset.
 
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  • #3
Is the question: Is [tex] [-1, \infty)[/tex] an open set?

Or is the question: Does [tex] [-1, \infty)[/tex] contain an open set?
 
  • #4
Office_Shredder said:
Is the question: Is [tex] [-1, \infty)[/tex] an open set?

Or is the question: Does [tex] [-1, \infty)[/tex] contain an open set?

What's the difference?
 
  • #5
It is not an open set. But it contains the open set (4, 7) for example.
 
  • #6
g_edgar said:
It is not an open set. But it contains the open set (4, 7) for example.
I guess I'm not understanding the OP's question. Any non zero interval on the reals "contains" every possible combination: [a,b],(a,b),(a,b],[a,b). Any such interval has a bijective mapping to the entire set of reals, so of course the interval [-1,[tex]\infty)[/tex] "contains" open sets.

EDIT: Perhaps I'm mistaken, but in terms of open and closed sets or subsets, I'm considering the actual membership of a given set to be dependent on the specification (choice) of that set. Therefore I could specify that every subset of C:C subset of R be closed.
 
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FAQ: Does the Interval [-1,∞) Include Any Open Sets?

What is an open set?

An open set is a subset of a mathematical space that does not contain any of its boundary points. In other words, all points within an open set are considered to be "interior" points.

How do you determine if a set is open?

To determine if a set is open, you can use the definition of an open set: if every point within the set has a neighborhood that is also contained within the set, then the set is considered open.

What is the difference between an open set and a closed set?

An open set does not contain any of its boundary points, while a closed set contains all of its boundary points. Additionally, a set can be both open and closed, known as a clopen set.

Can a set be both open and closed?

Yes, a set can be both open and closed. This type of set is known as a clopen set and is a property of certain mathematical spaces, such as topological spaces.

Why are open sets important in mathematics?

Open sets are important in mathematics because they allow for the definition of continuity and convergence, which are fundamental concepts in analysis and topology. They also play a key role in the definition of differentiable functions and the study of topological properties of spaces.

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