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# Why is [0,infinity) a closed set? Watch

1. [0,1] is closed, [0,1) is open/closed, so why is it that [0,infinity) is a closed set?

I thought if you include the upper limit the set is closed, but as there is no real number for infinity how is closed?
2. (Original post by BackToMathsAgain)
[0,1] is closed, [0,1) is open, so why is it that [0,infinity) is a closed set?

I thought if you include the upper limit the set is closed, but as there is no real number for infinity how is closed?
Is [0,1) not half open/half closed
[0,infinity) is half open/half closed

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3. (Original post by rayquaza17)
Is [0,1) not half open/half closed
[0,infinity) is half open/half closed

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my bad on [0,1), you are correct. My notes say that [0,infinity) is closed though :s
4. (Original post by BackToMathsAgain)
my bad on [0,1), you are correct. My notes say that [0,infinity) is closed though :s
That's weird: my notes say it's open, but when I google it people say it's closed?

Maybe someone with some more knowledge can help us.

EDIT: I've checked Thomas' Calculus (twelfth edition) and on ap-3 it says that [a,infinity) is a closed set. As does Wolfram MathWorld.

Wikipedia however (which obviously is a great source compared to the two mentioned above) says this notation is ambiguous?

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5. I've just checked my notes again, and the lecturer says we could class [0,infinity) as a closed set, but the explanation is beyond the scope of the module??

I'm in favour of this being ambiguous, but I'd like to hear what anyone else has to say??
6. In a topological sense, with the usual topology on , is a closed set because the complement is an open set (that is, for any you can find an such that )
7. I don't think you understand what it means for a set to be closed if you cannot see why [0,∞), with its usual topology, is closed. Reread your definition, it has absolutely nothing to do with brackets. Furthermore, [0,1) is not open/closed, it is neither.
8. The way I visualize this for the real line is to imagine a sequence that converges. A set S is closed only if every sequence that converges also converges to a point in S. So in other words, [0,1] is closed because if and then . Similarly, is closed because every convergent sequence will also have a limit in . Obviously is not closed because of .

Edit: I hope this isn't too confusing, but a set can be open and simultaneously closed. I won't go into details, but the point is that open and closed are just names. They are also stupid names, which is why something can be open and closed simultaneously.
9. It contains all of its own limit points. So it's closed.

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11. (Original post by IrrationalNumber)
The way I visualize this for the real line is to imagine a sequence that converges. A set S is closed only if every sequence that converges also converges to a point in S.
Shouldn't this say "A set S is closed only if every sequence of points in S that converges also converges to a point in S"?

By the way, it's clear that this is a very tricky topic.
12. (Original post by atsruser)
By the way, it's clear that this is a very tricky topic.
Wow, there really is a video like this for everything that upsets people! 😂

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