Finding Interior Points in a Subset: Hints & Tips

[mathanon]

How do you find the interior points of a subset?

I understand that a point is an interior point if there exists an epsilon neighborhood that is in the set, but I don't know how that would work with specific sets. Any hints?

 

[Jorriss]

If you have a set X and a subset of X called Y, and one wants to know if a point p is an interior point of Y, one need only find a neighborhood of p that is contained in Y.

For example, let X = R, and Y = Q (the rationals). Take an arbitrary point p in Q and take an epsilon neighborhood around p which is contained entirely in Q, that is, p is in (p - ε, p + ε). But it is known that every interval contains an irrational number, which contradicts our assumption that the prescribed interval is in Q. Therefore, as p was arbitrary, Q has no interior points.

Without being more specific to your needs, that is the best I can say.

 

[mathanon]

Thank you! That definitely helps!

 

[HallsofIvy]

Note that you titled this "isolation points" but asked about "interior points". An "isolated point" of a set cannot be an interior point.

For example, if A= (0, 1), the set of all x such that 0< x< 1, the interior points are just points in A itself. That is true because:
if x in (0, 1) then 0< x< 1. Let d1= x, d2= 1- x. If d1< d2, the neighborhood (x-d1, x+d1) is a subset of A. If d2< d1, (x-d2, x+ d2) is in A.

If A= [0, 1], the set of all x such that $0\le x\le 1$, the interior points are again the points in (0, 1). That's true because any neighborhood of "0", (-d, d), includes points outside A (negative numbers to -d) and any neighborhood of "1", (1-d, 1+ d), includes points outside A (numbers larger than 1 up to 1+ d) so "0" and "1", while in the set, are not interior[points].

Some other useful words: we say that point, p, is an "exterior" point of set A if and only if it is an interior point of the complement of A. The complement of (0, 1) is $$(-\infty, 0]cup [1, \infty)$$ and the complement of [0, 1] is $$(-\infty, 0)\cup (1, \infty)$$ both of which have $$(-\infty, 0)\cup(1, \infty)$$ as interior points (so that the "exterior" points of both (0, 1) and [0, 1] are $$(-\infty, 0)\cup (1, \infty)$$. The boundary points of a set are all points that are neither "interior points" nor "exterior points" of the set. Here, the boundary points of both (0, 1) and [0, 1] are the points "0" and "1".

The difference is that those boundary points are in [0, 1] and not in (0, 1). We say that (0, 1) containing none of its boundary points, is an "open" set and [0, 1], containing all of its boundary points, is a "closed" set.

Or course, a set may contain some of its boundary points but not all. (0, 1] is an example. Since neither "none of its boundary points" nor "all of its boundary points" is true, such a set is neither open nor closed.

Although it is unusual, it is possible for a set to have NO boundary points. In that case "none" and "all" are the same, such a set is both open and closed.

 

[blue_raver22]

There are a few techniques that can be used to find interior points in a subset. One approach is to first define the boundary of the subset, which is the set of points that are on the edge of the subset. This can be done using methods such as finding the closure or the boundary of the subset. Once the boundary is defined, you can then look for points within the subset that are not on the boundary. These points would be considered interior points.

Another approach is to use the definition of an interior point and check if a given point satisfies the condition. As you mentioned, a point is an interior point if there exists an epsilon neighborhood that is completely contained within the subset. This means that the point must be at least a distance of epsilon away from the boundary of the subset. By checking the distance of a point from the boundary of the subset, you can determine if it is an interior point.

Additionally, you can also use visual aids such as graphs or diagrams to help identify interior points. For example, if the subset is a geometric shape, you can plot it on a graph and visually identify points that are within the shape and not on the boundary.

Overall, finding interior points in a subset may require a combination of techniques and approaches, depending on the specific subset and its properties. It may also require some trial and error to determine the best method for a particular subset. I hope these hints and tips are helpful in guiding your search for interior points.