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Boundary in Mathematics | Think mathematically

Boundaries are controversial. Ask the politicians! Mathematicians, on the other hand, have an interesting way of thinking about the boundary of the space. It is less controversial for sure.

Pick any point in the space. Draw rays emanating from that point.

How do you make sure that you have drawn a ray?

Choose any two points on your drawing. Draw all paths connecting those two points. (You only have to work infinitely hard). Make sure the shortest of all these infinite paths lies in your drawing.

If this works for any pair of points on your drawing, then, voila, you have drawn a ray!

Have you drawn all rays emanating from a point? Excellent! You have found the boundary of your space.

Each ray is a point at the boundary!

Let us have a closer look at the core idea. What is the boundary of a space? It is a set of points associated with the space. This set of points (which we call the boundary) has some desirable properties. We can build this set of points in whatever way we want. As long as the final product (the final set) has desirable properties, we are happy.

Mathematicians think of the set of rays as the set of points in the boundary. This is clever! Why? This process yields a setbuilt out of the space we already have. We do not need to include external points that were not already inside our space to build the boundary. The wall is built using bricks from inside the space.

Mathematicians really love this type of constructions. (They are frugal in nature). They are always interested to do more with less. You hand them out some space and ask for its boundary points. If it is an infinite space (where you may keep on moving in some direction as long as you wish), a mathematician has played ner card and cooked up its set.

The real question is: does this set of rays (which we are calling boundary) has the desirable properties? More importantly, what properties do we desire from a point in the boundary of the space?

Naively speaking, a point at the boundary, should be the last point of the space. If you stand at a that point and move even a little, you are either inside the space or outside. Compare this to a non-boundary, interior point of the space. Moving in small enough steps from interior points you can stay inside the space. (You may have to ensure that you are moving by a really small amount).

Does the set of rays satisfy this property? How do you even compare moving from a set of rays to a set of points inside the space? This may lead to a technical description of the process, which I intended to avoid here.

However, the core idea is beautiful enough! Every infinite road is a boundary point. Even in an infinite space, you are always at the edge.

Note: This is a very loose description of the Gromov boundary of hyperbolic space. It is meant for the non-mathematical reader with little or no-technical knowledge in mathematics.

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