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Ends and tangles
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg ( IF 0.4 ) Pub Date : 2017-01-18 , DOI: 10.1007/s12188-016-0163-0
Reinhard Diestel

We show that an arbitrary infinite graph can be compactified by its $${\aleph _0}$$ℵ0-tangles in much the same way as the ends of a locally finite graph compactify it in its Freudenthal compactification. In general, the ends then appear as a subset of its $${\aleph _0}$$ℵ0-tangles. The $${\aleph _0}$$ℵ0-tangles of a graph are shown to form an inverse limit of the ultrafilters on the sets of components obtained by deleting a finite set of vertices. The $${\aleph _0}$$ℵ0-tangles that are ends are precisely the limits of principal ultrafilters.The $${\aleph _0}$$ℵ0-tangles that correspond to a highly connected part, or $${\aleph _0}$$ℵ0-block, of the graph are shown to be precisely those that are closed in the topological space of its finite-order separations.

中文翻译:

末端和缠结

我们表明,任意无限图可以通过其 $${\aleph _0}$$ℵ0 缠结来紧缩,其方式与局部有限图的末端在其 Freudenthal 紧缩中紧缩它的方式大致相同。通常,末端然后作为其 $${\aleph _0}$$ℵ0 缠结的子集出现。图中的 $${\aleph _0}$$ℵ0-缠结在通过删除有限顶点集获得的分量集上形成超滤器的逆极限。作为末端的$${\aleph _0}$$ℵ0-缠结正是主要超滤器的极限。$${\aleph _0}$$ℵ0-缠结对应于高度连接的部分,或$${\图的 aleph _0}$$ℵ0-block 恰好是在其有限阶分离的拓扑空间中封闭的那些。
更新日期:2017-01-18
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