Ends of a space

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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  such that ,

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 $\left[( (i, \infty) )_{i\in\mathbb{N}}\right], \left[((-\infty, -i))_{i\in\mathbb{N}}\right]$ .

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Categories: General topology | Properties of topological spaces

Ends of a space

From Wikipedia, the free encyclopedia

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