Set closure and interior points

In summary, for a set A that is a subset of a topological space X, the interior of A is the largest open set contained in A. In the case of A being the set of rational numbers between 0 and 1, the interior is the empty set. The closure of A is the smallest closed set containing A, which for this example is the interval [0, 1]. However, the interior of the closure of A is (0, 1), which is not the empty set. This is a good example that shows how the interior and closure of a set can differ.
  • #1
PBRMEASAP
191
2
If [tex]A \subset X[/tex] where [tex]X[/tex] has a topology, is it generally true that the interior of [tex]A[/tex] is equal to the interior of the closure of [tex]A[/tex]? This seems very reasonable to me, but probably only because I'm visualizing [tex]A[/tex] as a disc in the real plane. If it isn't true, what would be a counterexample?

thanks
 
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  • #2
Take X to be the set of real numbers with the usual topology.

Let A be the set of all rational numbers between 0 and 1 (inclusive).

What is the interior of A? What is the closure of A?

What is the interior of the closure of A?
 
  • #3
HallsofIvy said:
Take X to be the set of real numbers with the usual topology.

Let A be the set of all rational numbers between 0 and 1 (inclusive).

What is the interior of A? What is the closure of A?

What is the interior of the closure of A?

Okay this is a good example! Let me see if I have it straight. The interior of A is the largest open set contained in A, which would be the empty set in this case. I say this because the set has no interior points, or points for which a neighborhood can be found containing only points in A. The closure of A is the smallest closed set containing A, which is the interval [0, 1]. I say this because for every point in [0, 1], every neighborhood of that point contains points in A. But the interior of [0, 1] is (0, 1), which is not the empty set.

Did I understand correctly? I am a little foggy on the properties of real numbers, so I can't really back up my claims about the density of rationals at the moment.

Thanks for your help!
 
  • #4
yes...you've got it right...
 

1. What is set closure?

The set closure, also known as the closure or the topological closure, of a set is the set of all points that can be obtained by taking limits of points in the set. In other words, it is the smallest closed set that contains all the points in the original set. It is denoted by the symbol 🡇.

2. What is the difference between set closure and set interior?

The set interior, also known as the interior or the topological interior, of a set is the largest open set contained within the original set. It is denoted by the symbol 🡈. The main difference between set closure and set interior is that the set closure includes all the limit points of the set, while the set interior only includes points that are completely surrounded by the set.

3. How do you determine if a point is in the closure of a set?

A point is in the closure of a set if and only if every open set containing that point also contains points from the original set. In other words, the point is either in the set itself or is a limit point of the set. If the point is not in the closure, then it is not a limit point of the set and there exists an open set containing the point that does not contain any points from the original set.

4. Can a set have empty interior but non-empty closure?

Yes, it is possible for a set to have an empty interior but a non-empty closure. This means that there are no points completely surrounded by the set, but there are still points that are limit points of the set. An example of such a set is the set of all rational numbers, which has an empty interior but its closure is the set of all real numbers.

5. How are set closure and set interior related?

The set closure and set interior are complementary concepts. This means that the closure of a set is equal to the complement of the interior of the complement of the set. In other words, the closure of a set is the set of all points that are either in the set or on its boundary, while the interior of the set is the set of all points that are completely surrounded by the set.

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