End (topology)

In mathematics, the ends of a topological space are spoken clearly the connected components of the " edge at infinity ". Formally, they are defined as equivalence classes of complements of compact sets.

Definition

Be a topological space. We consider the family of compact subsets and for each of the set of connected components of.

On

We define an equivalence relation by

(In words: one connected component of is equivalent to belong to a connected component of if there is a compact set, so that and the same connected component of. )

The equivalence classes of the equivalence relation on hot ends of the topological space.

As environments an end the open sets are referred to the respective equivalence class.

Examples

• The number line has two ends.
• For having an end.
• Be the interior of a compact manifold with boundary, ie. Then the ends of the corresponding connected components of.

• Be the Cayley graph of a free group. Then infinitely many ends has, there is a bijection of the ends on a Cantor set.
• By a theorem of Freudenthal has the Cayley graph of a group is either infinitely many or at most two ends.