- #1

- 4

- 0

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?

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter mathanon
- Start date

- #1

- 4

- 0

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?

- #2

- 1,082

- 25

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.

- #3

- 4

- 0

Thank you! That definitely helps!

- #4

HallsofIvy

Science Advisor

Homework Helper

- 41,833

- 963

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 [itex]0\le x\le 1[/itex], the interior points are again the points in (0, 1). That's true because

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

The difference is that those boundary points are

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

Share: