Question about open sets in (-infinite,5]

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In summary: All points with distance smaller then 1 are in B(5,1) but, this means of course all point that are in your space. Otherwise it wouldn't make much sence.
  • #1
dodo
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The stupid question of the day.

If S is the real interval (-infinite, 5], and I can find a metric d so that (S,d) is a metric space, then,

is, for example, (4, 5] an open set in (S,d) ?

I say this because, the way I'm reading the definition of an open ball, the open ball B(5,1) is the interval (4,5] and not the interval (4,6), since the points in (5,6) do not belong to the metric space (S,d). So every open ball in (4,5] centered in 5 is completely contained in (4,5].
 
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  • #2
Yes, (4,5] is open.
 
  • #3
Thanks, micromass, just checking the fundamentals.

By the way, I apologize for the phrasing of the question; if the metric d is not specifically defined, then there is really no way to tell. As someone else pointed me out, the question should have referred to the restriction of the Euclidean metric to the set S.
 
  • #4
In that case you automatically have to deal with the quotient topology on S and obviously (4,5] is certainly the intersection of an open set of R and S. Of course just using the definition of metric also works. All points with distance smaller then 1 are in B(5,1) but, this means of course all point that are in your space. Otherwise it wouldn't make much sence.
 
  • #5
micromass said:
Yes, (4,5] is open.

Not necessarily. The question posed by the OP is if he can find a metric turning [itex](-\infty, 5] [/itex] into a metric space, then is (4,5] necessarily open. This is the question as it's posed, and the answer is not necessarily. He did not specify what the metric would be, so there's no guarantee that the metric would in any way resemble the regular Euclidean metric. There are many metrics that one can construct on [itex]\mathbb{R}[/itex], and then a restriction to his subspace S yields a metric on S. In some of those metrics, (4,5] may be open. In others, they may not.

If the question were specifically about the regular Euclidean metric, then the answer is yes.
 
  • #6
... read third post
 
  • #7
Thanks all for your answers! My actual doubt was about open balls with the Euclidean metric, but I did a awful job formulating it -- it's clear now.
 
  • #8
conquest said:
In that case you automatically have to deal with the quotient topology on S and obviously (4,5] is certainly the intersection of an open set of R and S. Of course just using the definition of metric also works. All points with distance smaller then 1 are in B(5,1) but, this means of course all point that are in your space. Otherwise it wouldn't make much sence.

The quotient topology, as far as I learned, deals with an equivalent relation on the space and is defined on the space of equivalent classes of the original space. This case is just a subspace defined by the intersection of the open sets with the subset. Correct me if I'm wrong.
 

1. What is an open set?

An open set in mathematics is a set of numbers or points on a number line that does not include its endpoints. In other words, every point in the set has a neighborhood that is also contained within the set.

2. 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 in the set has a neighborhood that is also contained within the set, then the set is open. You can also check if the set does not include its endpoints.

3. Why is the interval (-infinite,5] not considered an open set?

The interval (-infinite,5] is not considered an open set because it includes its endpoint at 5. A point at the endpoint does not have a neighborhood contained within the set, violating the definition of an open set.

4. Can a set be both open and closed?

Yes, a set can be both open and closed. This type of set is called a clopen set. An example of a clopen set is the empty set, which does not include any points and is both open and closed.

5. How are open sets used in real-world applications?

Open sets are used in real-world applications in fields such as physics, engineering, and economics. They are particularly useful in modeling continuous systems and analyzing their properties. For example, in physics, open sets are used to represent the possible values of a continuous variable such as position or energy.

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