End (topology)

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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 X - K in a compatible way, so that

${\displaystyle K_{1}\subseteq K_{2}\Rightarrow e(K_{1})\supseteq e(K_{2}).\,}$

If X is compact, then there are no ends.

Examples[edit]

• The real line 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 K: 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 has one end, ∞. If K is a compact, hence closed and bounded, subset of the plane, contained in the disc of radius r, say, then there is a single unbounded component to X-K, containing the complement of the disc.

Compactification[edit]

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