Star (topology)

In the mathematical theory of the simplicial complexes , a star is a certain environment of a simplex.

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  Two simplices and their degree.

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  A corner and its star.

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  A corner and its link.

Let be a simplicial complex and a set of simplices in . ${\displaystyle K}$ ${\displaystyle S}$ ${\displaystyle K}$

The conclusion of is the smallest sub-simplicial complex of that contains. So it contains and all pages of simplizes in . ${\displaystyle S}$ ${\displaystyle K}$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle S}$

The (open) star of a simplex is the union of all simplices that contain as a side. The (open) star of is the union of the stars of all simplices from . A closed star is the closure of the star of . ${\displaystyle \sigma }$ ${\displaystyle \sigma }$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle S}$

The link from is the difference between the termination of the star of and the star of the termination of . ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle S}$

literature

Ralph Stöcker, Heiner Zieschang : Algebraic Topology. Teubner, Stuttgart 1988, ISBN 3-519-02226-5