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In Algebraic topology, Čech homology theory is a theory named after the mathematician Eduard Čech. It is based on the idea of using open sets in topological spaces as the vertices of simplicial complexes.

Čech homology itself is the limit of simplicial homology groups of the Čech nerve.

Čech complex

Čech homology groups
Let $$X $$ be a topological space

A Čech homology system is a collection of homomorphisms

Then Čech homology groups over any over any coefficient group$$G$$ are the inverse limit of the $$q^{\text {th }}$$ Čech homology system of the pair $$(X, A)$$ over $$G$$.

It is denoted by $$$H_q(X, A ; G)$ $\left[H^{\mathrm{q}}(X, A ; G)\right]$$$.

Given a topological space X,  Čech homology is the limit of simplicial homology-groups of the Čech nerves of its open covers, under refinement of open covers.

The Cech homology theory is defined on the category of arbitrary pairs $(X, A)$ and their maps. The coefficient group $G$ is taken to be an $\mathrm{R}$-module for any ring $R$, and the resulting $H^{\circ}(X, A)$ are in $\varrho_R$.

Relation to Čech cohomology
Norman Steenrod constructed Čech cohomology by dualizing Čech homology in 1936.

Čech homology is related to Čech cohomology by an isomorphism.

Comparison to other homologies
Čech homology theory is not a homology theory in the sense of Eilenberg-Steenrod as it does not fulfill the Exactness Axiom. It does, however, fulfill additional axioms proposed by Milnor and is, compared to singular homology, highly responsive to the space's shape.

For compact metric spaces, there is an isomorphism of the Čech and Vietoris homology groups.