Exploring Open Cover of Interval [0,1)

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In summary, the conversation discusses different options for an open cover with infinite subcovers for the interval [0,1). The participants consider using sets of the form An = (-1/n, n) with n \in lN and also explore the idea of staying within the boundaries of the set being covered. They conclude that neither (-1,1) nor (-n,n) are examples of covers without a finite subcover, but (-1,1-1/n) and (-1,1-(1/n)) are both valid options.
  • #1
Bachelier
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Given the interval [0,1)

is this a good Open cover with infinite subcovers

{(An)} such that An =(-1/n, n) with n [tex]\in[/tex] lN

Is there any reason we should stay to the boundaries of the set we're trying to cover?

I'm thinking that even (-n, n) should work.

Am I wrong?
 
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  • #2
I think I just answered my own question.

I think to make sure that the portion that covers the interval is infinite.

Hence (-1, 1 - (1/n)) would work.
 
  • #3
Bachelier said:
Given the interval [0,1)

is this a good Open cover with infinite subcovers

{(An)} such that An =(-1/n, n) with n [tex]\in[/tex] lN

Is there any reason we should stay to the boundaries of the set we're trying to cover?

I'm thinking that even (-n, n) should work.

Am I wrong?

Neither of these is an example of a cover without a finite subcover. In the first case, [itex](-1,1) \supset [0,1)[/itex]; same in the second case.
 
  • #4
AxiomOfChoice said:
Neither of these is an example of a cover without a finite subcover. In the first case, [itex](-1,1) \supset [0,1)[/itex]; same in the second case.

Agreed, but [itex](-1,1-1/n)[/itex]
is.
 
  • #5
Bachelier said:
I think I just answered my own question.

I think to make sure that the portion that covers the interval is infinite.

Hence (-1, 1 - (1/n)) would work.

Yep. No finite subcovers here!
 

1. What is an open cover of interval [0,1)?

An open cover of interval [0,1) is a collection of open sets that completely cover the interval [0,1) without including its endpoints. In other words, every point in [0,1) is contained in at least one of the open sets in the cover.

2. Why is exploring open cover of interval [0,1) important in mathematics?

Exploring open cover of interval [0,1) is important in mathematics because it allows us to study and understand the properties of the interval [0,1) in a more detailed and systematic way. It also has applications in various fields, such as topology, analysis, and geometry.

3. How do you construct an open cover of interval [0,1)?

An open cover of interval [0,1) can be constructed by choosing a collection of open sets, such as intervals or half-open intervals, that cover all the points in [0,1) without including the endpoints. One common example is the cover of [0,1) by the open intervals (1/n, 1-1/n) for all positive integers n.

4. Can there be more than one open cover of interval [0,1)?

Yes, there can be infinitely many open covers of interval [0,1). For example, in addition to the cover mentioned in question 3, we can also have the cover of [0,1) by the open intervals (1/(n+1), 1-1/(n+1)) for all positive integers n. Both covers completely cover [0,1) without including its endpoints.

5. How is the concept of open cover related to the concept of compactness?

The concept of open cover is closely related to the concept of compactness. In fact, a set is compact if and only if every open cover of the set has a finite subcover. This means that for any open cover of a compact set, we can choose a finite number of open sets from the cover that still cover the entire set. In the case of interval [0,1), the set is not compact because there exist open covers that do not have a finite subcover, such as the cover mentioned in question 4.

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