Relative Boundaries in General Topology

In summary, the conversation discusses the problem of proving that the boundary of a set U relative to A is a subset of the intersection of A and the boundary of U relative to X, where X is a topological space with A as a subset. The conversation also explains the concept of the "relative topology" and how it relates to the boundary of a set. The conversation concludes with a suggestion to use the given definition to prove the result.
  • #1
quantumdude
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Hi,

I was trying to help a student with an assignment in topology when I was stumped by a symbol that I had not seen before. Here's the problem.

a.) Let [itex](X,\square)[/itex] be a topological space with [itex]A\subseteq X[/itex] and [itex]U\subseteq A[/itex]. Prove that [itex]Bd_A(U)\subseteq A\cap Bd_X(U)[/itex].

The first thing that has got me stumped here is the subscripted boundaries. I have never seen this before, but I tried to reason it out as follows. The "ordinary" boundary of a set A is [itex]Bd(A)=[ext(A)]^c\cap[int(A)]^c[/itex], the intersection of all the points that are neither in the exterior of [itex]A[/itex] nor in the interior of [itex]A[/itex]. The first problem is how to relate the boundary of a set to a second set (and thus introduce the subscripts), so I went back to the definition of the complement of a set [itex]A[/itex], which is the difference [itex]\mathbb{U}-A[/itex], where [itex]\mathbb{U}[/itex] is the universal set. This led me to conjecture that:

[itex]Bd_A(U)=[A-ext(U)]\cap[A-int(U)][/itex]
[itex]Bd_X(U)=[X-ext(U)]\cap[X-int(U)][/itex]

Before I move on, can someone tell me if that is correct? Thanks.
 
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  • #2
Yes, [tex]Bd_A(U) is the "boundary of U relative to A" which means the boundary of U in the "relative topology". If X is a topological space and A is a subset of A, then all open sets "relative to A" are open sets in X, intersect A. The interior of U "relative to A" is simply the interior of U (relative to X) intersect A and exterior of U "relative to A" is the exterior of U (relative to X) intersect A. Since boundary points of U are points of U that are neither interior nor exterior to U, but points in the boundary of U "relative to A" must be in A, they must be neither interior nor exterior "relative to X": that is must be in the boundary of U "relative to X": in boundary of U relative to X intersect A, just as the formla says.
 
  • #3
http://www.ornl.gov/sci/ortep/topology/defs.txt

contains some definitions.

As an example, consider [0,1] as a subset of R: it's boundary os {0,1}, but in the subspace topology its boundary is empty.

If you use the definition there it should become clear how to prove this result.
 
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Related to Relative Boundaries in General Topology

1. What is the definition of relative boundaries in general topology?

Relative boundaries in general topology refer to the boundaries of a subset of a topological space. It is defined as the set of points that are shared by the subset and its complement.

2. Why are relative boundaries important in topology?

Relative boundaries play a crucial role in topology as they help define the topological properties of a subset. They also aid in understanding the relationship between the subset and the entire space.

3. How are relative boundaries different from absolute boundaries in topology?

Absolute boundaries refer to the boundaries of a set in a topological space, while relative boundaries pertain to the boundaries of a subset within that space. In other words, relative boundaries take into account the context of the subset within the larger space.

4. Can relative boundaries have different properties than absolute boundaries?

Yes, relative boundaries can have different properties than absolute boundaries. This is because the relative boundary takes into account both the subset and its complement, while the absolute boundary only considers the set itself. This can lead to different topological properties and behaviors.

5. How are relative boundaries used in practice?

Relative boundaries are used in many applications, such as data analysis and image processing, to identify boundaries and distinguish between different regions. They are also used in mathematical proofs and constructions in topology.

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