Relative Interior, Interior & Boundary: Answers to Your Questions

[kaosAD]

I am confused with the terms Relative Interior of a set and Interior of a set. Can someone enlighten me. Also, there is a term Relative Boundary. What does this relative signify?

 

[fourier jr]

this is only a guess, so i could easily be wrong. anytime you have a set Y in a topological space X, which is also a topological space you say that Y is a subspace of X, with the relative topology. relative interior could refer to the interiors of subsets of Y, which also happen to be interiors of subsets of X. to get these all you do is for A subset of X find int(A intersect Y) to get the relative interior. so int(A) would be the interior of A in X and int(A intersect Y) is the (relative) interior of A in Y, even though they could very well be the same set. relative boundary would be similar i would think. that's only an educated guess & nothing else, as I've never seen those terms before.

 

[HallsofIvy]

Relative to what?

For example, if (0, 1] is a half-open interval of the real numbers, its interior is the open set (0,1) and its boundary is the set {0, 1}.
However, it boundary relative to the set (0, 2] is just {1} because 0 is not contained in (0,2] and we are considering only points in (0,2]

The interior of (0,1] relative to (0, 2] is still (0, 1) but its interior relative to, say, (-1, 1] is (0, 1]. 1 is now an interior point "relative to" (-1, 1] because we are now considering (-1, 1] as "everything there is"- the basic set for the topology. and interval about 1 with radius, say, 1/4 is completely contained in (0, 1]. The fact that 1+ 1/8 is outside that interval doesn't matter- it is outside (-1, 1] also and so doesn't count.

 

[kaosAD]

Thank you for the insight. It is very helpful. I have asked this question in another forum and I got the the same question in turn -- relative to what?

I thought the definition of relative interior is standard, but I might be wrong. I found in a book which defines relative interior of a set A is an interior of A relative to affine hull of A.

I wonder what relative interior is useful for.

 

[HallsofIvy]

Thank you for the insight. It is very helpful. I have asked this question in another forum and I got the the same question in turn -- relative to what?
I thought the definition of relative interior is standard, but I might be wrong. I found in a book which defines relative interior of a set A is an interior of A relative to affine hull of A.
I wonder what relative interior is useful for.

It is pretty much the nature of the word "relative" that it must be "relative" to something!

If we have several different sets, so that each is a subset of another, say A subset B subset C, then I can think of A as being a subset of C, ignoring B, or think of A as a subset of B, ignoring C. If we are given a topology on C, that is, a collection of open sets, then B has the "relative topology"- each open set in B is one of the opens sets of C intersect B.

Given that, A may be an open subset of B but not of C: A would be open "relative to B" but not "relative to C". Similarly, the interior of A as a subset of B (interior relative to B) might be different than the interior of A thought of as a subset of C (interior relative to C). That might become important if you have a function that is defined only for some of the points of C.

 

[kaosAD]

Thank you for the insight.