End (topology)

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In topology, a branch of mathematics, an end of a topological space is a point in a certain kind of compactification of the space.

Let X be a non-compact topological space. Suppose that K is a non-empty compact subset of X, and $V \subseteq X\backslash K$ a connected component of $X\backslash K$ , and V ⊆ U ⊆ X an open set containing V. Then U is a neighborhood of an end of X.

An end of X is an equivalence class of sequences $X \supset U_1 \supset U_2 \supset \cdots$ such that $\cap \overline{U}_i = \varnothing$ , where $U i$ is a neighborhood of an end.

Two such sequences $(U i),(V j)$ are equivalent if for all i, there exists j such that $U_i \supset V_j$ , and for all j, there exists i such that $V_j \supset U_i$ . Given an end $\mathcal{E}$ and a neighborhood of an end $U$ , $U$ is called a neighborhood of $\mathcal{E}$ if there is a sequence $(U i)$ such that $[(U_i)]=\mathcal{E}$ and $U_1 \subset U$ . The notion of an end of a topological space was introduced by Hans Freudenthal.

For example, $\mathbb{R}$ has two ends, with ends given by $\left[( (n, \infty) )_{n\in\mathbb{N}}\right], \left[((-\infty, -n))_{n\in\mathbb{N}}\right]$ .

Ends can be characterized in a number of ways using algebraic functors. For example, the set of compact subsets of $X$ is partially ordered by inclusion. Taking complements defines a partial order on the set of complements $X - K$ where $K$ ranges over all compact sets. An inclusion $K \to L$ of compact sets induces a map, using the $π 0$ functor, from $X-L \to X-K$ . The inverse limit