# Vietoris homology

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An (ordered) -dimensional -simplex of a subset of a metric space is defined as an ordered subset in subject to the condition . The -chains of are then defined for a given coefficient group as formal finite linear combinations of -simplices with coefficients . The boundary of an -simplex is defined as follows: ; this is an -chain. By linearity, the boundary of any -chain is defined and -cycles are defined as -chains with zero boundary. An -chain of a set is -homologous to zero in (the notation is ) if for a certain -chain in .
A true cycle of a set is a sequence in which is an -cycle in and ( ). The true cycles form a group, . A true cycle is homologous to zero in if for any there exists an such that all for are -homologous to zero in . One denotes by the quotient group of the group by the subgroup of cycles that are homologous to zero.
A cycle is called convergent if for any there exists an such that any two cycles , are mutually -homologous in if . The group of convergent cycles is denoted by . Let be the corresponding quotient group.
A cycle has compact support if there exists a compact set such that all the vertices of all simplices of all cycles lie in . One similarly modifies the concept of a cycle being homologous to zero by requiring the presence of a compact set on which all the homology-realizing chains lie; convergent cycles with compact support can thus be defined. By denoting with a subscript the transition to cycles and homology with compact support, one obtains the groups and . The latter group is known as the Vietoris homology group. If the polyhedron is finite, the Vietoris homology groups coincide with the standard homology groups.
Relative homology groups , , , modulo a subset are also defined. An -cycle of the set modulo is any -chain in for which the chain lies in . In a similar manner, an -cycle modulo is -homologous modulo to zero in if , where and are -chains in , while the chain lies in .