In his Algebraic Topology book, Hatcher defines the chain groups for simplicial homology as free abelian groups with basis the open $n$-simplices of some simplicial complex X. Is there any benefit/difference by using open $n$-simplices in comparison to using closed $n$-simplices?
2026-04-29 23:07:00.1777504020
Simplicial homology: chain group with basis open n-simplices vs. chain group with basis closed n-simplices
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I expect the restriction to open $n$-simplices is just to remove any potential ambiguity from the fact that closed $n$-simplices may overlap, while their interiors do not.