End (topology)

In general topology, an end of a topological space generalises the notion of "point at infinity" of the real line or plane.

An end of a topological space X is a function e which assigns to each compact set K in X some connected component with non-compact closure e(K) of the complement XK in a compatible way, so that

$K_1 \subseteq K_2 \Rightarrow e(K_1) \supseteq e(K_2) .\,$

If X is compact, then there are no ends.

Examples

• The real line R has two ends, which may be denoted ±∞. If K is a compact subset of R then by the Heine-Borel theorem K is closed and bounded. There are two unbounded components of RK: if K is contained in the interval [a,b], they are the components containing (-∞,a) and (b,+∞). An end is a consistent choice of the left- or the right-hand component.
• The real plane R2 has one end, ∞. If K is a compact, hence closed and bounded, subset of the plane, contained in a disk, then there is a single unbounded component of R2K, the component containing the complement of the disc.

Compactification

Denote the set of ends of X by E(X) and let $X^* = X \cup E(X)$. We may topologise X * by taking as neighbourhoods of an end e the sets $N_K(e) = e(K) \cup \{f \in E(X) : f(K)=e(K) \}$ for compact K in X.

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