Question about open sets in [0,1]

AxiomOfChoice · 2010-04-10T20:21:45+00:00

Isn't [0,1) open in [0,1]? I know it's not open in \mathbb R, but I sincerely hope it's open in [0,1].

Thread starter AxiomOfChoice
Start date Apr 10, 2010
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Sets

In summary, an open set in [0,1] is a set of numbers that includes all of the numbers between 0 and 1, but does not include the endpoints 0 and 1. A set in [0,1] is open if every point in the set has a neighborhood that is completely contained within the set. An open set in [0,1] does not include its endpoints, while a closed set includes its endpoints. Open sets are important in mathematics because they are a fundamental concept in topology, allowing for the definition of topological concepts such as continuity and convergence. An example of an open set in [0,1] is (0,1/2), which includes all numbers between 0 and

Apr 10, 2010

AxiomOfChoice

Isn't [0,1) open in [0,1]? I know it's not open in [itex]\mathbb R[/itex], but I sincerely hope it's open in [0,1].

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Apr 10, 2010

Hurkyl

Staff Emeritus

Science Advisor

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Yes, it is. The easiest way to prove it is probably to find an open set in R whose intersection with [0,1] is [0,1).

What is an open set in [0,1]?

An open set in [0,1] is a set of numbers that includes all of the numbers between 0 and 1, but does not include the endpoints 0 and 1.

How do you determine if a set is open in [0,1]?

A set in [0,1] is open if every point in the set has a neighborhood that is completely contained within the set. In other words, for any point in the set, there is a small enough interval around that point that is also entirely within the set.

What is the difference between an open set and a closed set in [0,1]?

An open set in [0,1] does not include its endpoints, while a closed set includes its endpoints. This means that a closed set in [0,1] would include the numbers 0 and 1, while an open set would not.

Why are open sets important in mathematics?

Open sets are important in mathematics because they are a fundamental concept in topology, which is the study of the properties of spaces and the relationships between them. Open sets allow for the definition of topological concepts such as continuity and convergence.

Can you give an example of an open set in [0,1]?

One example of an open set in [0,1] is the set (0,1/2), which includes all numbers between 0 and 1/2, but does not include 0 or 1/2. This set is open because any point within the set, such as 0.25, has a neighborhood (in this case, the interval (0.2, 0.3)) that is completely contained within the set.