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 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.

[edit] Examples

[edit] 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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