In mathematics, the Gromov boundary of a δ-hyperbolic space (especially a hyperbolic group) is an abstract concept generalizing the boundary sphere of hyperbolic space. Conceptually, the Gromov boundary is the set of all points at infinity. For instance, the Gromov boundary of the real line is two points, corresponding to positive and negative infinity.
Suppose X is any set. It is, Suppose, we have defined a distance function or a metric on this set.
What is a distance function?
It is a function d takes pairs of elements of X as input and gives away the distance (a nonnegative real number) between them as an output. This distance or metric d is usually computed by weird formulas. We only require the metric d to satisfy the following properties:
- d(x, y) > 0 is x and y are different (for all members x, y from X)
- d(x, x) = 0
- d(x, y) = d(y, x)
- \( d(x, y) + d(y, z) \ge d(x, z) \)
If you think closely, all of these properties mimic the notion of distance that we are familiar with. The last one is the triangular inequality.
Once we have defined a metric d on the set X, we can define some subsets of X as open sets. First, we define open balls. An open ball centered at ( x_o \in X ) and of radius ( \epsilon ) is the set of all points in X which are less then (\epsilon ) distance away from ( x_0 ). This distance is of course measured by the distance function that we defined earlier.
- Open Set: U is an open set in X, if for every \( x \in U \), it is possible to have an open ball containing x that is contained in U.
- Closed Set: V is a closed set if \( V^c \) is open in X.
- Bounded Set: W is a bounded set if there exists a finite number M, such that distance between any pair of members of X is at most M.
Sometimes we need to relax the notion of ‘closed and bounded’ sets. Imagine you are ‘covering’ a subset U with open sets. Intuitively speaking, think about the open sets as carpets. You are covering U with this collection of ‘carpets’ mean, that U is contained in the union of this collection of open sets (usually containing infinitely many ‘carpets’).
A set is said to be compact if whenever you can cover it with infinitely many carpets, you will be able to cover it using finitely many carpets of that collection. In the usual n-dimensional Euclidean space, closed and bounded is compact (this needs proof).
Collection of all open sets defines a topology on the set X (yes, the topology on X is just a collection of subsets of the set X, which have the designation of being ‘open’; and this ‘openness’ is defined as above).
Now we have a set X, a distance function (metric) d and a topology induced by it. This apparatus (the set, along with metric and topology induced by it) is known as a metric space. Suppose that this metric space is proper. This means, that every closed and bounded set is compact (hence it has some similarity with the usual Euclidean space that we are familiar with).